Introduction
Discrete diffusion models have emerged as a compelling alternative to autoregressive models. By allowing multiple positions to be updated in parallel, they offer a path toward faster decoding [23 , 26 ] and potentially more efficient steering and control [29 ] . In continuous time, these models are described by forward and reverse continuous-time Markov chains [3 , 23 ] . Standard simulation of the reverse process is inherently stochastic: Gillespie’s algorithm draws fresh randomness at each jump [14 ] , while ancestral sampling of discrete-time reverse chains likewise relies on stochastic transitions [30 , 26 , 19 , 29 ] .
This raises a natural question: can one design a discrete diffusion sampler whose only randomness is the initial condition, as in the probability-flow ODE for continuous diffusion models [32 a] ? A direct analogue on a finite state space is impossible because a deterministic map cannot split probability mass. Continuous auxiliary randomness is therefore required at initialization, particularly for masked diffusion, whose terminal distribution is a Dirac mass. Existing uses of auxiliary randomness serve different purposes. The projected Gaussian trajectories of Sahoo et al. [27 ] act as more of a reparameterization and do not define a deterministic flow driven solely by the current discrete state. Gumbel Distillation (Zhang et al., 2026) instead uses Gumbel noise to construct deterministic training targets from an autoregressive teacher for parallel decoding, rather than a marginal-preserving deterministic flow for a discrete diffusion process.
Our contributions are as follows:
Our central contribution is a new deterministic sampler for discrete diffusion models, which we call the exponential residual flow (ERF). We augment the state with non-negative auxiliary variables, called residuals, initialized as independent exponential random variables, one for each token and position. Their evolution is then governed by an ODE determined by the forward and reverse rates of the discrete dynamics that governed jumps of the discrete state. A token change is triggered whenever a residual reaches zero. In contrast to Gillespie’s first-reaction construction, where residual clocks are refreshed after each jump, ERF continues to evolve all remaining residuals deterministically. Consequently, once the initial augmented state has been sampled, no additional randomness is required. We prove that these augmented dynamics preserve the marginal distribution of the original CTMC at every time. At each fixed time, the residuals remain independent standard exponential random variables and are independent of the discrete state. The resulting process generally differs from the reverse Markov chain and need not be Markovian.
To make this construction applicable to learned diffusion models, we introduce a novel numerical integration scheme that freezes the leave-one-out denoiser over each sampling interval while preserving the known time dependence of the transition rates. We derive explicit formulations for both masked and uniform diffusion models, and show that the resulting scheme is exact for factorized data distributions when the denoiser is exact.
We evaluate ERF against ancestral sampling under matched model-evaluation budgets and find comparable generation quality. The deterministic nature of ERF also makes it directly amenable to consistency distillation. We exploit this property to train few-step student models and evaluate the resulting speed–quality trade-offs on text generation, code generation, and Sudoku completion.
Notation. We consider integer-valued sequences in X = V L X = V^L X = V L V = ⟦ 1 , K ⟧ V = \llbracket1, K \rrbracket V = [ [ 1 , K ] ] Δ K = { p ∈ R K : p k ≥ 0 , ∑ k = 1 K p k = 1 } \Delta_K = \{p \in\mathbb{R}^K : p_k \ge0,\ \sum_{k=1}^{K} p^k = 1\} Δ K = { p ∈ R K : p k ≥ 0 , ∑ k = 1 K p k = 1 } x ∈ X x \in X x ∈ X x ℓ x^\ell x ℓ ℓ \ell ℓ x − ℓ ∈ V L − 1 x^{-\ell} \in V^{L-1} x − ℓ ∈ V L − 1 ℓ \ell ℓ m : = K m := K m := K x = ( x 1 , … , x d ) ∈ V d x = (x^1,\ldots,x^d) \in V^d x = ( x 1 , … , x d ) ∈ V d x ~ ∈ { 0 , 1 } d × K \tilde{x} \in\{0,1\}^{d \times K} x ~ ∈ { 0 , 1 } d × K x ~ ℓ k : = 1 x ℓ = k \tilde{x}^{\ell k} := \mathbf{1}_{x^\ell=k} x ~ ℓ k := 1 x ℓ = k x ∈ V x \in V x ∈ V x ~ ∈ R K \tilde{x} \in\mathbb{R}^K x ~ ∈ R K π ∈ Δ K \pi\in\Delta_K π ∈ Δ K Cat ( π ) \operatorname{Cat}(\pi) Cat ( π ) Cat ( x ; π ) : = ⟨ x ~ , π ⟩ \operatorname{Cat}(x;\pi) := \langle\tilde{x},\pi\rangle Cat ( x ; π ) := ⟨ x ~ , π ⟩ x x x Cat ( π ) \operatorname{Cat}(\pi) Cat ( π ) sup ∅ = − ∞ \sup\varnothing= -\infty sup ∅ = − ∞
Background on discrete diffusion models
We first briefly introduce discrete diffusion models [26 , 30 , 2 , 19 , 29 ] . This class of generative models constructs a stochastic process that transports a simple reference distribution, p 1 : = ∏ ℓ = 1 L Cat ( π ℓ ) p_1 := \prod_{\ell=1}^{L} \operatorname{Cat}(\pi^\ell) p 1 := ∏ ℓ = 1 L Cat ( π ℓ ) π ∈ ( Δ K ) L \pi\in(\Delta_K)^L π ∈ ( Δ K ) L p 0 p_0 p 0 p 1 p_1 p 1
We consider as forward process a Continuous Time Markov Chain (CTMC) process ( Y t ) t ∈ [ 0 , 1 ] (Y_t)_{t \in[0,1]} ( Y t ) t ∈ [ 0 , 1 ] p 0 p_0 p 0 q t ∣ s ( x t ∣ x s ) = ∏ ℓ = 1 L q t ∣ s ℓ ( x t ℓ ∣ x s ℓ ) q_{t|s}(x_t|x_s) = \prod_{\ell=1}^{L} q_{t|s}^{\ell}(x_t^\ell|x_s^\ell) q t ∣ s ( x t ∣ x s ) = ∏ ℓ = 1 L q t ∣ s ℓ ( x t ℓ ∣ x s ℓ ) x s , x t ∈ X x_s,x_t \in X x s , x t ∈ X 0 ≤ s < t ≤ 1 0 \le s < t \le1 0 ≤ s < t ≤ 1
q t ∣ s ℓ ( x t ℓ ∣ x s ℓ ) = Cat ( x t ℓ ; α t ∣ s x ~ s ℓ + ( 1 − α t ∣ s ) π ℓ ) , 0 ≤ s < t ≤ 1 , (1) q_{t|s}^{\ell}(x_t^\ell|x_s^\ell) = \operatorname{Cat}(x_t^\ell; \alpha_{t|s}\tilde{x}_s^\ell+ (1-\alpha_{t|s})\pi^\ell), \qquad0 \le s < t \le1 , \tag*{(1)} q t ∣ s ℓ ( x t ℓ ∣ x s ℓ ) = Cat ( x t ℓ ; α t ∣ s x ~ s ℓ + ( 1 − α t ∣ s ) π ℓ ) , 0 ≤ s < t ≤ 1 , (1) where x ~ s \tilde{x}_s x ~ s x s x_s x s ( α t ) t ∈ [ 0 , 1 ] (\alpha_t)_{t\in[0,1]} ( α t ) t ∈ [ 0 , 1 ] α 0 = 1 \alpha_0 = 1 α 0 = 1 α 1 ≈ 0 \alpha_1 \approx0 α 1 ≈ 0 α t ∣ s = α t / α s \alpha_{t|s} = \alpha_t/\alpha_s α t ∣ s = α t / α s ( Y 1 − t ) t ∈ [ 0 , 1 ] (Y_{1-t})_{t\in[0,1]} ( Y 1 − t ) t ∈ [ 0 , 1 ] 1 Y t = lim h ↓ 0 Y t − h Y_t = \lim_{h\downarrow0} Y_{t-h} Y t = lim h ↓ 0 Y t − h Y t Y_t Y t X t X_t X t Y t Y_t Y t t t t X t X_t X t p t p_t p t p 1 p_1 p 1 p 0 p_0 p 0 ( X t ) t ∈ [ 0 , 1 ] (X_t)_{t\in[0,1]} ( X t ) t ∈ [ 0 , 1 ]
p s ∣ t ( x s ∣ x t ) = ∑ x 0 q s ∣ 0 , t ( x s ∣ x 0 , x t ) p 0 ∣ t ( x 0 ∣ x t ) , (2) p_{s|t}(x_s|x_t) = \sum_{x_0} q_{s|0,t}(x_s|x_0,x_t)\,p_{0|t}(x_0|x_t) , \tag*{(2)} p s ∣ t ( x s ∣ x t ) = x 0 ∑ q s ∣0 , t ( x s ∣ x 0 , x t ) p 0∣ t ( x 0 ∣ x t ) , (2) with p 0 ∣ t ( ⋅ ∣ x t ) p_{0|t}(\cdot|x_t) p 0∣ t ( ⋅ ∣ x t ) q s ∣ 0 , t q_{s|0,t} q s ∣0 , t Y s Y_s Y s ( Y 0 , Y t ) (Y_0,Y_t) ( Y 0 , Y t ) § 11 . During training, these reverse transitions are learned using a factorized model parameterized by θ \theta θ p s ∣ t θ ( x s ∣ x t ) = ∏ ℓ = 1 L p s ∣ t θ ( x s ℓ ∣ x t ) p^\theta_{s|t}(x_s|x_t) = \prod_{\ell=1}^{L} p^\theta_{s|t}(x_s^\ell|x_t) p s ∣ t θ ( x s ∣ x t ) = ∏ ℓ = 1 L p s ∣ t θ ( x s ℓ ∣ x t )
p 0 : n θ ( x t 0 : n ) = p 1 ( x t n ) ∏ i = 1 n p t i − 1 ∣ t i θ ( x t i − 1 ∣ x t i ) , (3) p^\theta_{0:n}(x_{t_0:n}) = p_1(x_{t_n}) \prod_{i=1}^{n} p^\theta_{t_{i-1}|t_i}(x_{t_{i-1}}|x_{t_i}) , \tag*{(3)} p 0 : n θ ( x t 0 : n ) = p 1 ( x t n ) i = 1 ∏ n p t i − 1 ∣ t i θ ( x t i − 1 ∣ x t i ) , (3) for a choice of timesteps 0 = t 0 < ⋯ < t n = 1 0 = t_0 < \cdots< t_n = 1 0 = t 0 < ⋯ < t n = 1 s i = t i − 1 s_i = t_{i-1} s i = t i − 1 [30 , 26 , 19 , 29 ] ) is p s ∣ t θ ( x s ∣ x t ) = q s ∣ 0 , t ( x s ∣ x ^ 0 θ ( x t , t ) , x t ) p^\theta_{s|t}(x_s|x_t) = q_{s|0,t}(x_s|\hat{x}^\theta_0(x_t,t),x_t) p s ∣ t θ ( x s ∣ x t ) = q s ∣0 , t ( x s ∣ x ^ 0 θ ( x t , t ) , x t ) q s ∣ 0 , t q_{s|0,t} q s ∣0 , t (17) to simplex-valued inputs, and x ^ 0 θ \hat{x}^\theta_0 x ^ 0 θ ( Δ K ) L (\Delta_K)^L ( Δ K ) L ℓ \ell ℓ ( t , x t ) ↦ E [ X 0 ℓ ∣ X t − ℓ = x t − ℓ ] (t,x_t) \mapsto\mathbb{E}[X^\ell_0 \mid X^{-\ell}_t = x^{-\ell}_t] ( t , x t ) ↦ E [ X 0 ℓ ∣ X t − ℓ = x t − ℓ ] § 11 .
Rate matrices and discrete score. Another useful perspective on discrete diffusion is provided through rate matrices associated to the forward and backward processes ([3 , 23 ] (Sun et al. 2023)). The rate matrix associated to the forward transitions (1) is given for any x , y ∈ X x,y \in\mathcal{X} x , y ∈ X t ∈ [ 0 , 1 ] t \in[0,1] t ∈ [ 0 , 1 ]
Q t ( y ∣ x ) = ∑ ℓ = 1 L 1 y − ℓ = x − ℓ Q t ℓ ( y ℓ ∣ x ℓ ) , where Q t ℓ ( y ∣ x ) = ( − α ˙ t α t ) ⟨ y ~ , π ℓ ⟩ ( x ≠ y ) . (4) Q_t(y|x) = \sum_{\ell=1}^{L} \mathbf{1}_{y^{-\ell}=x^{-\ell}}Q^\ell_t(y^\ell|x^\ell), \qquad\text{where } Q^\ell_t(y|x)=\left(-\frac{\dot{\alpha}_t}{\alpha_t}\right)\langle\tilde{y},\pi^\ell\rangle\quad(x\ne y). \tag*{(4)} Q t ( y ∣ x ) = ℓ = 1 ∑ L 1 y − ℓ = x − ℓ Q t ℓ ( y ℓ ∣ x ℓ ) , where Q t ℓ ( y ∣ x ) = ( − α t α ˙ t ) ⟨ y ~ , π ℓ ⟩ ( x = y ) . (4) Here x , y ∈ X x,y \in\mathcal{X} x , y ∈ X Q t ℓ ( x ∣ x ) = − ∑ y ≠ x Q t ℓ ( y ∣ x ) Q^\ell_t(x|x)=-\sum_{y\ne x}Q^\ell_t(y|x) Q t ℓ ( x ∣ x ) = − ∑ y = x Q t ℓ ( y ∣ x ) ℓ ∈ [ 1 , L ] \ell\in[1,L] ℓ ∈ [ 1 , L ] ( p t ) t ∈ [ 0 , 1 ] (p_t)_{t\in[0,1]} ( p t ) t ∈ [ 0 , 1 ] ( Y t ) t ∈ [ 0 , 1 ] (Y_t)_{t\in[0,1]} ( Y t ) t ∈ [ 0 , 1 ] (1) and initial distribution p 0 p_0 p 0 ( p t ) t ∈ [ 0 , 1 ] (p_t)_{t\in[0,1]} ( p t ) t ∈ [ 0 , 1 ] [11 ] ): for t ∈ ( 0 , 1 ) t\in(0,1) t ∈ ( 0 , 1 ) x , y ∈ X x,y\in\mathcal{X} x , y ∈ X
∂ t p t ( y ) = ∑ x ∈ X Q t ( y ∣ x ) p t ( x ) . (5) \partial_t p_t(y)=\sum_{x\in\mathcal{X}}Q_t(y|x)p_t(x). \tag*{(5)} ∂ t p t ( y ) = x ∈ X ∑ Q t ( y ∣ x ) p t ( x ) . (5) Note that the forward transition q t ∣ s ( ⋅ ∣ x s ) q_{t|s}(\cdot|x_s) q t ∣ s ( ⋅ ∣ x s ) (1) are obtained by solving (5) started from δ x s \delta_{x_s} δ x s s s s ( Q ‾ t ) t ∈ [ 0 , 1 ] (\overline{Q}_t)_{t\in[0,1]} ( Q t ) t ∈ [ 0 , 1 ] 2 Q ‾ t \overline{Q}_t Q t y ≠ x y\ne x y = x p t ( x ) > 0 p_t(x)>0 p t ( x ) > 0 3
Q ‾ t ( y ∣ x ) : = p t ( y ) p t ( x ) Q t ( x ∣ y ) = ∑ ℓ = 1 L 1 y − ℓ = x − ℓ p t ℓ ( y ℓ ∣ x − ℓ ) p t ℓ ( x ℓ ∣ x − ℓ ) Q t ℓ ( x ℓ ∣ y ℓ ) . (6) \overline{Q}_t(y|x):=\frac{p_t(y)}{p_t(x)}Q_t(x|y)=\sum_{\ell=1}^{L}\mathbf{1}_{y^{-\ell}=x^{-\ell}}\frac{p^\ell_t(y^\ell|x^{-\ell})}{p^\ell_t(x^\ell|x^{-\ell})}Q^\ell_t(x^\ell|y^\ell). \tag*{(6)} Q t ( y ∣ x ) := p t ( x ) p t ( y ) Q t ( x ∣ y ) = ℓ = 1 ∑ L 1 y − ℓ = x − ℓ p t ℓ ( x ℓ ∣ x − ℓ ) p t ℓ ( y ℓ ∣ x − ℓ ) Q t ℓ ( x ℓ ∣ y ℓ ) . (6) Algorithm 1 Gillespie next-reaction / ERF algorithm.
Require: 0 ≤ s ≤ t ≤ 1 0 \le s \le t \le1 0 ≤ s ≤ t ≤ 1 ( Q ‾ u ) u ∈ [ s , t ] (\overline{Q}_u)_{u\in[s,t]} ( Q u ) u ∈ [ s , t ] ( λ u ) u ∈ [ s , t ] (\lambda_u)_{u\in[s,t]} ( λ u ) u ∈ [ s , t ]
Output: ( X s , E s ) (X_s,E_s) ( X s , E s ) s s s
Initialize: ( X t , E t ) ∼ p t ⊗ Exp ⊗ ( L × K ) (X_t,E_t) \sim p_t \otimes\operatorname{Exp}^{\otimes(L\times K)} ( X t , E t ) ∼ p t ⊗ Exp ⊗ ( L × K ) X s ← X t X_s \leftarrow X_t X s ← X t τ ⋆ ← t \tau^\star\leftarrow t τ ⋆ ← t E s ← E t E_s \leftarrow E_t E s ← E t
while τ ⋆ > s \tau^\star> s τ ⋆ > s do
τ ℓ j ← sup { u ∈ [ s , τ ⋆ ] : ∫ u τ ⋆ Q ‾ r ℓ ( j ∣ X s ) d r ≥ E s ℓ j } \tau^{\ell j} \leftarrow\sup\left\{u \in[s,\tau^\star] : \int_u^{\tau^\star} \overline{Q}^{\ell}_r(j\mid X_s)\,\mathrm{d}r \ge E_s^{\ell j}\right\} τ ℓ j ← sup { u ∈ [ s , τ ⋆ ] : ∫ u τ ⋆ Q r ℓ ( j ∣ X s ) d r ≥ E s ℓ j } j ≠ X s ℓ j \ne X_s^\ell j = X s ℓ
( ℓ ⋆ , j ⋆ ) ← argmax ( ℓ , j ) : j ≠ X s ℓ τ ℓ j (\ell^\star,j^\star) \leftarrow\operatorname*{argmax}_{(\ell,j):j\ne X_s^\ell}\tau^{\ell j} ( ℓ ⋆ , j ⋆ ) ← argmax ( ℓ , j ) : j = X s ℓ τ ℓ j
τ n e w ⋆ ← max { s , τ ℓ ⋆ j ⋆ } \tau^\star_{\mathrm{new}} \leftarrow\max\{s,\tau^{\ell^\star j^\star}\} τ new ⋆ ← max { s , τ ℓ ⋆ j ⋆ }
E s ℓ j ← E s ℓ j − ∫ τ n e w ⋆ τ ⋆ Q ‾ r ℓ ( j ∣ X s ) d r E_s^{\ell j} \leftarrow E_s^{\ell j} - \int_{\tau^\star_{\mathrm{new}}}^{\tau^\star}\overline{Q}^{\ell}_r(j\mid X_s)\,\mathrm{d}r E s ℓ j ← E s ℓ j − ∫ τ new ⋆ τ ⋆ Q r ℓ ( j ∣ X s ) d r j ≠ X s ℓ j \ne X_s^\ell j = X s ℓ
if X s ℓ = j X_s^\ell= j X s ℓ = j then ▷ update passive residuals
either E s ℓ j ∼ Exp ( 1 ) E_s^{\ell j} \sim\operatorname{Exp}(1) E s ℓ j ∼ Exp ( 1 )
or E s ℓ j ← E s ℓ j + ∫ τ n e w ⋆ τ ⋆ λ r ℓ ( j ) d r E_s^{\ell j} \leftarrow E_s^{\ell j} + \int_{\tau^\star_{\mathrm{new}}}^{\tau^\star}\lambda_r^\ell(j)\,\mathrm{d}r E s ℓ j ← E s ℓ j + ∫ τ new ⋆ τ ⋆ λ r ℓ ( j ) d r
end if
if τ ℓ ⋆ j ⋆ ≥ s \tau^{\ell^\star j^\star} \ge s τ ℓ ⋆ j ⋆ ≥ s then
X s ℓ ⋆ ← j ⋆ X_s^{\ell^\star} \leftarrow j^\star X s ℓ ⋆ ← j ⋆
end if
τ ⋆ ← τ n e w ⋆ \tau^\star\leftarrow\tau^\star_{\mathrm{new}} τ ⋆ ← τ new ⋆
end while
return X s , E s X_s,E_s X s , E s
with Q ‾ t ( x ∣ x ) : = λ ‾ t ( x ) = − ∑ y ≠ x Q ‾ t ( y ∣ x ) \overline{Q}_t(x\mid x) := \overline{\lambda}_t(x) = -\sum_{y\ne x}\overline{Q}_t(y\mid x) Q t ( x ∣ x ) := λ t ( x ) = − ∑ y = x Q t ( y ∣ x ) (2) , (5) and (6) , we obtain that the reverse transitions solve the following Kolmogorov equation: for any x t , y ∈ X x_t,y \in\mathcal{X} x t , y ∈ X p t ( x t ) > 0 p_t(x_t)>0 p t ( x t ) > 0
∂ s p s ∣ t ( y ∣ x t ) = − ∑ x Q ‾ s ( y ∣ x ) p s ∣ t ( x ∣ x t ) , p t ∣ t ( y ∣ x t ) = 1 x t = y . (7) \partial_s p_{s\mid t}(y\mid x_t) = -\sum_x \overline{Q}_s(y\mid x)p_{s\mid t}(x\mid x_t), \qquad p_{t\mid t}(y\mid x_t) = \mathbb{1}_{x_t=y}. \tag*{(7)} ∂ s p s ∣ t ( y ∣ x t ) = − x ∑ Q s ( y ∣ x ) p s ∣ t ( x ∣ x t ) , p t ∣ t ( y ∣ x t ) = 1 x t = y . (7) Deterministic sampling via exponential residual flows
In this section, we construct a sampler whose trajectory is deterministic after a single random initialization and with one-time marginals agreeing with those of the reverse CTMC. The key idea is to augment the discrete state with the exponential variables used in Gillespie’s first-reaction method and replace their resampling at jump times with a deterministic continuous evolution, resulting in the Exponential Residual Flow sampler described in Algorithm 1.
Exponential Residual Flow
The CTMC ( X t ) t ∈ [ 0 , 1 ] (X_t)_{t\in[0,1]} ( X t ) t ∈ [ 0 , 1 ] X t X_t X t X t X_t X t
Gillespie’s algorithm. Gillespie’s global algorithm (Gillespie, [14 ] ) constructs the piecewise constant trajectory of the reverse CTMC one jump at a time. Starting from X t = x X_t=x X t = x τ < t \tau<t τ < t X u = x X_u=x X u = x u ∈ ( τ , t ] u\in(\tau,t] u ∈ ( τ , t ] E ∼ Exp ( 1 ) E\sim\operatorname{Exp}(1) E ∼ Exp ( 1 ) ∫ τ t λ ‾ u ( x ) d u = E \int_\tau^t\overline{\lambda}_u(x)\,\mathrm{d}u=E ∫ τ t λ u ( x ) d u = E τ \tau τ ℓ ∈ [ L ] \ell\in[L] ℓ ∈ [ L ] j ≠ x ℓ j\ne x^\ell j = x ℓ Q ‾ τ ℓ ( j ∣ x ) / λ ‾ τ ( x ) \overline{Q}^{\ell}_\tau(j\mid x)/\overline{\lambda}_\tau(x) Q τ ℓ ( j ∣ x ) / λ τ ( x ) X τ ℓ = j X^\ell_\tau=j X τ ℓ = j τ \tau τ X τ X_\tau X τ § 12 .§ 1 for the formal construction.
Note that this construction has two sources of randomness: the exponential draw determining the jump time and the categorical draw determining the token change. Our goal is to eliminate randomness beyond initialization. We now describe Gillespie’s first-reaction algorithm , which removes the need for the categorical draw (Gillespie, [14 ] ; Anderson, [1 ] ). Given X t = x X_t=x X t = x E ℓ j ∼ Exp ( 1 ) E^{\ell j}\sim\operatorname{Exp}(1) E ℓ j ∼ Exp ( 1 ) ℓ ∈ [ L ] \ell\in[L] ℓ ∈ [ L ] j ≠ x ℓ j\ne x^\ell j = x ℓ
∫ τ ℓ j t Q ‾ u ℓ ( j ∣ x ) d u = E ℓ j . (8) \int_{\tau^{\ell j}}^{t} \overline{Q}^{\ell}_{u}(j|x)\,\mathrm{d}u = E^{\ell j}. \tag*{(8)} ∫ τ ℓ j t Q u ℓ ( j ∣ x ) d u = E ℓ j . (8) The next jump encountered backward from t t t ( ℓ ⋆ , j ⋆ ) ∈ argmax ( ℓ , j ) : j ≠ x ℓ τ ℓ j (\ell^\star, j^\star) \in\operatorname*{argmax}_{(\ell,j):j\ne x^\ell} \tau^{\ell j} ( ℓ ⋆ , j ⋆ ) ∈ argmax ( ℓ , j ) : j = x ℓ τ ℓ j τ = τ ℓ ⋆ j ⋆ \tau= \tau^{\ell^\star j^\star} τ = τ ℓ ⋆ j ⋆ x x x ( τ , t ] (\tau,t] ( τ , t ] τ \tau τ X τ ℓ ⋆ = j ⋆ X_{\tau}^{\ell^\star}=j^\star X τ ℓ ⋆ = j ⋆ τ \tau τ X τ X_\tau X τ L ( K − 1 ) L(K-1) L ( K − 1 )
Next-reaction method. The next-reaction method reuses the exponential variables associated with token changes that were not selected ([13 , 1 , 8 ] ). To see how, we can express the calculation of the next jump as a continuous evolution. Given X t = x X_t=x X t = x E t ℓ j = E ℓ j E_t^{\ell j}=E^{\ell j} E t ℓ j = E ℓ j j ≠ x ℓ j\ne x^\ell j = x ℓ
d E u ℓ j d u = Q ‾ u ℓ ( j ∣ x ) , \frac{\mathrm{d}E_u^{\ell j}}{\mathrm{d}u}=\overline{Q}^{\ell}_{u}(j|x), d u d E u ℓ j = Q u ℓ ( j ∣ x ) , with time evolving backwards. We call these evolving variables residuals. The first residual to reach zero determines the jump time τ \tau τ ( ℓ ⋆ , j ⋆ ) (\ell^\star,j^\star) ( ℓ ⋆ , j ⋆ ) (8) for every pair and selecting the largest candidate time in line 4. Throughout this evolution, the discrete state stays equal to x x x τ \tau τ X τ ℓ ⋆ = j ⋆ X_{\tau}^{\ell^\star}=j^\star X τ ℓ ⋆ = j ⋆
We use a variant of this construction, given in Algorithm 1, that maintains one residual for every position–token pair, initialized in line 1. Residuals with j ≠ x ℓ j\ne x^\ell j = x ℓ active and are updated in line 6. Those with j = x ℓ j=x^\ell j = x ℓ passive and are resampled in line 8. This variant requires L L L
Exponential Residual Flow. The exponential residual flow (ERF) eliminates this remaining randomness by keeping the evolution of the active residuals unchanged and replacing the resampling of the L L L ( X 1 , E 1 ) ∼ p 1 ⊗ Exp ( 1 ) ⊗ ( L × K ) (X_1,E_1)\sim p_1\otimes\operatorname{Exp}(1)^{\otimes(L\times K)} ( X 1 , E 1 ) ∼ p 1 ⊗ Exp ( 1 ) ⊗ ( L × K ) ( X 1 , E 1 ) (X_1,E_1) ( X 1 , E 1 ) E t ℓ j E_t^{\ell j} E t ℓ j j = x t ℓ j=x_t^\ell j = x t ℓ
We can deduce from the algorithm a process ( X t , E t ) t ∈ [ 0 , 1 ] (X_t,E_t)_{t\in[0,1]} ( X t , E t ) t ∈ [ 0 , 1 ] [ 0 , 1 ] [0,1] [ 0 , 1 ] ( τ k ) k ≥ 0 (\tau_k)_{k\ge0} ( τ k ) k ≥ 0 k k k ( τ k + 1 , τ k ] (\tau_{k+1},\tau_k] ( τ k + 1 , τ k ] X t X_t X t X τ k X_{\tau_k} X τ k E t E_t E t
d d t E t ℓ j = { Q ‾ t ℓ ( j ∣ X t ) , j ≠ X t ℓ , − λ t ℓ ( X t ℓ ) , j = X t ℓ . (9) \frac{\mathrm{d}}{\mathrm{d}t}E_t^{\ell j} =
\begin{cases}
\overline{Q}^{\ell}_t(j|X_t), & j\ne X_t^\ell,\\
-\lambda^\ell_t(X_t^\ell), & j=X_t^\ell.
\end{cases} \tag*{(9)} d t d E t ℓ j = { Q t ℓ ( j ∣ X t ) , − λ t ℓ ( X t ℓ ) , j = X t ℓ , j = X t ℓ . (9) We call the process ( X t , E t ) t ∈ [ 0 , 1 ] (X_t,E_t)_{t\in[0,1]} ( X t , E t ) t ∈ [ 0 , 1 ] Figure 1 .
Figure 1. The Exponential Residual Flow algorithm. Evolution of the joint state ( X t , E t ) (X_t,E_t) ( X t , E t ) X 1 ∼ p 1 X_1 \sim p_1 X 1 ∼ p 1 E 1 ∼ Exp ( 1 ) ⊗ ( 3 × 6 ) E_1 \sim\operatorname{Exp}(1)^{\otimes(3 \times6)} E 1 ∼ Exp ( 1 ) ⊗ ( 3 × 6 ) t t t X t X_t X t
We show in our next result that the resulting process ( X t ) t ≥ 0 (X_t)_{t\ge0} ( X t ) t ≥ 0
Theorem 3.1 (Correctness of the exponential residual flow). Under standard assumptions stated in Appendix I the ERF process is almost surely well defined with finitely many jumps, and for every t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ]
L ( X t , E t ) = p t ⊗ Exp ( 1 ) ⊗ ( L × K ) . (10) \mathcal{L}(X_t,E_t)=p_t\otimes\operatorname{Exp}(1)^{\otimes(L\times K)}. \tag*{(10)} L ( X t , E t ) = p t ⊗ Exp ( 1 ) ⊗ ( L × K ) . (10) To the best of our knowledge, Theorem 3.1 yields a novel deterministic procedure for sampling the marginals of a CTMC which is deterministic given the initialization, similarly to the probability flow ODE in the continuous case ([31 ] ).
We motivate the choice of the velocity (9) in Appendix § 18 .§ 1 by deriving a continuity equation on the extended state space from the Kolmogorov equation (5) satisfied by ( p t ) t ∈ [ 0 , 1 ] (p_t)_{t\in[0,1]} ( p t ) t ∈ [ 0 , 1 ] (9) . This argument, however, does not by itself prove Theorem 3.1 . A complete proof instead characterizes by induction the density of the process at successive jump times and decomposes the path space according to the number of jumps. We give this proof in Appendix § 18 .
Integrating the exponential residual flow
We now describe the practical implementation of Algorithm 1. For y ≠ x y \ne x y = x (6) are commonly approximated as
Q ˉ t θ ( y ∣ x ) : = ∑ ℓ = 1 L 1 y − ℓ = x − ℓ ⟨ y ~ ℓ , s t θ ( x ) ℓ ⟩ Q t ℓ ( x ℓ ∣ y ℓ ) , (11) \bar{Q}^{\theta}_{t}(y|x) := \sum_{\ell=1}^{L} \mathbf{1}_{y^{-\ell}=x^{-\ell}} \left\langle\tilde{y}^{\ell},s^{\theta}_{t}(x)^{\ell}\right\rangle Q^{\ell}_{t}(x^{\ell}|y^{\ell}), \tag*{(11)} Q ˉ t θ ( y ∣ x ) := ℓ = 1 ∑ L 1 y − ℓ = x − ℓ ⟨ y ~ ℓ , s t θ ( x ) ℓ ⟩ Q t ℓ ( x ℓ ∣ y ℓ ) , (11) where the score approximation s t θ ( x ) ℓ s^{\theta}_{t}(x)^{\ell} s t θ ( x ) ℓ (6) [23 ] . We must then estimate the integrals of these approximate reverse rates in lines 3-6 and those of the associated forward exit rates in line 9. However, exact integration may be computationally challenging or infeasible.
To address this problem, most samplers introduce discretization strategies for these integration steps and consider a grid 0 = t 0 < ⋯ < t n = 1 0=t_0<\cdots<t_n=1 0 = t 0 < ⋯ < t n = 1 τ \tau τ [ s i , t i ] [s_i,t_i] [ s i , t i ] X t i X_{t_i} X t i ( X u ) u ∈ [ s i , t i ] (X_u)_{u\in[s_i,t_i]} ( X u ) u ∈ [ s i , t i ] Q ˉ u θ ( y ; t i ∣ x ) = Q ˉ t i θ ( y ∣ X t i ) \bar{Q}^{\theta}_{u}(y;t_i|x)=\bar{Q}^{\theta}_{t_i}(y|X_{t_i}) Q ˉ u θ ( y ; t i ∣ x ) = Q ˉ t i θ ( y ∣ X t i ) § 12 .§ 4 . Another option, given the form of the reverse rate and its approximation (11) , is to freeze only the score approximation at time t i t_i t i Q u ℓ ( y ℓ ∣ x ℓ ) Q^\ell_u(y^\ell|x^\ell) Q u ℓ ( y ℓ ∣ x ℓ ) (4) , whose time integrals are available in closed form, similarly to Conforti et al. [6 ] . We develop a different alternative here. First, note that the score is related to the leave-one-out denoiser by
⟨ y ~ ℓ , s t θ ( x t ) ℓ ⟩ = q t ∣ 0 ( y ℓ ∣ x ^ 0 θ ( x t , t ) ℓ ) q t ∣ 0 ( x ℓ ∣ x ^ 0 θ ( x t , t ) ℓ ) . (12) \left\langle\tilde{y}^{\ell},s^{\theta}_{t}(x_t)^{\ell}\right\rangle=\frac{q_{t|0}\left(y^{\ell}\mid\hat{x}^{\theta}_{0}(x_t,t)^{\ell}\right)}{q_{t|0}\left(x^{\ell}\mid\hat{x}^{\theta}_{0}(x_t,t)^{\ell}\right)}. \tag*{(12)} ⟨ y ~ ℓ , s t θ ( x t ) ℓ ⟩ = q t ∣0 ( x ℓ ∣ x ^ 0 θ ( x t , t ) ℓ ) q t ∣0 ( y ℓ ∣ x ^ 0 θ ( x t , t ) ℓ ) . (12) This relation suggests to freeze only the leave-one-out denoiser at time t i t_i t i (12) in (11) , the integration with respect to time of the resulting approximate reverse rate can be made explicit for most models of interest. This choice has the advantage that it yields an exact sampler for a single token or a factorized distribution when the leave-one-out predictor is optimal contrary to other freezing strategies; see Appendix § 12 .5.2.
Practical implementation. Given ( X ^ t i , E ^ t i ) (\hat{X}_{t_i},\hat{E}_{t_i}) ( X ^ t i , E ^ t i ) ( X ^ s i , E ^ s i ) (\hat{X}_{s_i},\hat{E}_{s_i}) ( X ^ s i , E ^ s i ) [ s i , t i ] [s_i,t_i] [ s i , t i ]
Q ˉ r θ ( y ; t i ∣ x ) : = ∑ ℓ = 1 L 1 y − ℓ = x − ℓ q r ∣ 0 ℓ ( y ℓ ∣ x ^ 0 θ ( X ^ t i , t i ) ℓ ) q r ∣ 0 ℓ ( x ℓ ∣ x ^ 0 θ ( X ^ t i , t i ) ℓ ) Q r ℓ ( x ℓ ∣ y ℓ ) , r ∈ [ s i , t i ] . (13) \bar{Q}^{\theta}_{r}(y;t_i|x):=\sum_{\ell=1}^{L}\mathbf{1}_{y^{-\ell}=x^{-\ell}}\frac{q^{\ell}_{r|0}\left(y^{\ell}\mid\hat{x}^{\theta}_{0}(\hat{X}_{t_i},t_i)^{\ell}\right)}{q^{\ell}_{r|0}\left(x^{\ell}\mid\hat{x}^{\theta}_{0}(\hat{X}_{t_i},t_i)^{\ell}\right)}Q^{\ell}_{r}(x^{\ell}|y^{\ell}),\qquad r\in[s_i,t_i]. \tag*{(13)} Q ˉ r θ ( y ; t i ∣ x ) := ℓ = 1 ∑ L 1 y − ℓ = x − ℓ q r ∣0 ℓ ( x ℓ ∣ x ^ 0 θ ( X ^ t i , t i ) ℓ ) q r ∣0 ℓ ( y ℓ ∣ x ^ 0 θ ( X ^ t i , t i ) ℓ ) Q r ℓ ( x ℓ ∣ y ℓ ) , r ∈ [ s i , t i ] . (13) For the forward exit rates, we keep the original rates ( λ r ℓ ) r ∈ [ s i , t i ] (\lambda^\ell_r)_{r\in[s_i,t_i]} ( λ r ℓ ) r ∈ [ s i , t i ] [ s i , t i ] [s_i,t_i] [ s i , t i ] (4) , can lead to infinitely many jumps. For leave-one-out freezing, however, (13) is the exact time reversal, on [ s i , t i ] [s_i,t_i] [ s i , t i ] x ^ 0 θ ( X ^ t i , t i ) \hat{x}^{\theta}_{0}(\hat{X}_{t_i},t_i) x ^ 0 θ ( X ^ t i , t i ) § 12 .§ 5 .
Specialization to MDM and UDM. Having specified the partially frozen reverse rate matrix in (13) , we now specify the calculation of jump times in line 3-4 of Algorithm 1 to MDM and UDM. For a current state X r X_r X r r r r r ∈ [ s i , t i ] r\in[s_i,t_i] r ∈ [ s i , t i ] E r ℓ E^\ell_r E r ℓ ∫ u r Q ˉ v θ , ℓ ( j ; t i ∣ X r ) d v = E r ℓ \int_u^r \bar{Q}^{\theta,\ell}_v(j;t_i|X_r)\,\mathrm{d}v=E^\ell_r ∫ u r Q ˉ v θ , ℓ ( j ; t i ∣ X r ) d v = E r ℓ u u u X r ℓ = m X^\ell_r=m X r ℓ = m
∫ u r Q ˉ v θ , ℓ ( j ; t i ∣ z ) d v = x ^ 0 θ ( X t i , t i ) j log ( 1 − α r 1 − α u ) (14) \int_u^r \bar{Q}^{\theta,\ell}_v(j;t_i|z)\,\mathrm{d}v=\hat{x}^{\theta}_{0}(X_{t_i},t_i)^j\log\left(\frac{1-\alpha_r}{1-\alpha_u}\right) \tag*{(14)} ∫ u r Q ˉ v θ , ℓ ( j ; t i ∣ z ) d v = x ^ 0 θ ( X t i , t i ) j log ( 1 − α u 1 − α r ) (14) and its candidate time follows by solving for u u u u u u k k k § 12 .§ 5 . To reduce both the number of jumps and the cost of computing their times, we exploit the fact that different rate matrices can generate the same marginals ( p t ) t ∈ [ 0 , 1 ] (p_t)_{t\in[0,1]} ( p t ) t ∈ [ 0 , 1 ] [4 , 3 , 12 ] ) and use the net-flux reverse rates ((Zhang et al., 2023)), defined for x ≠ y x\ne y x = y
Q ˉ t N F ( y ∣ x ) : = [ p t ( x ) Q ˉ t ( y ∣ x ) − p t ( y ) Q ˉ t ( x ∣ y ) ] + p t ( x ) = − ∑ ℓ = 1 L 1 y − ℓ = x − ℓ α ˙ t K α t [ p t ( y ℓ ∣ x t − ℓ ) p t ( x ℓ ∣ x t − ℓ ) − 1 ] + (15) \bar{Q}^{\mathrm{NF}}_t(y\mid x):=\frac{\left[p_t(x)\bar{Q}_t(y\mid x)-p_t(y)\bar{Q}_t(x\mid y)\right]_+}{p_t(x)}=-\sum_{\ell=1}^{L}\mathbf{1}_{y^{-\ell}=x^{-\ell}}\frac{\dot{\alpha}_t}{K\alpha_t}\left[\frac{p_t(y^\ell\mid x_t^{-\ell})}{p_t(x^\ell\mid x_t^{-\ell})}-1\right]_+ \tag*{(15)} Q ˉ t NF ( y ∣ x ) := p t ( x ) [ p t ( x ) Q ˉ t ( y ∣ x ) − p t ( y ) Q ˉ t ( x ∣ y ) ] + = − ℓ = 1 ∑ L 1 y − ℓ = x − ℓ K α t α ˙ t [ p t ( x ℓ ∣ x t − ℓ ) p t ( y ℓ ∣ x t − ℓ ) − 1 ] + (15) where a + = max { a , 0 } a_+=\max\{a,0\} a + = max { a , 0 } Q ˉ t \bar{Q}_t Q ˉ t § 10 , with the net-flux construction in Appendix § 13 .
Parallel sampling. Algorithm 1 processes jumps sequentially across positions because a jump may change the reverse rates. However, for MDM, UDM, and Uniform Net Flux with leave-one-out freezing, the frozen reverse rate at position ℓ \ell ℓ Q ˉ t ℓ ( y ∣ z ) \bar{Q}^{\ell}_t(y\mid z) Q ˉ t ℓ ( y ∣ z ) z z z z ℓ z^\ell z ℓ § 10 and § 12 .§ 5 . This means that changes at other positions than ℓ \ell ℓ ℓ \ell ℓ [ s i , t i ] [s_i,t_i] [ s i , t i ] L L L
Application to distillation
As in the continuous diffusion case ([28 , 33 ] ), one application of deterministic sampling is to distill a teacher diffusion model into a student model that performs better with few sampling steps. The absence of a direct analogue of the probability flow ODE in the discrete setting has often been used as an argument against trajectory-based distillation ([27 , 7 ] (Zhang et al. 2023)). Our exponential residual flow supplies deterministic transport between augmented states ( X t , E t ) (X_t,E_t) ( X t , E t ) ( X s , E s ) (X_s,E_s) ( X s , E s )
Consistency distillation. We rely on consistency distillation (CD) following Song et al. [33 ] . From an augmented state, the student predicts the clean discrete state at time 0 0 0 Ψ s ∣ t ϕ \Psi^\phi_{s|t} Ψ s ∣ t ϕ ( X s , E s ) = Ψ s ∣ t ϕ ( X t , E t ) (X_s,E_s)=\Psi^\phi_{s|t}(X_t,E_t) ( X s , E s ) = Ψ s ∣ t ϕ ( X t , E t ) s < t s<t s < t [27 ] couple X s X_s X s X t X_t X t X t X_t X t § 14 .§ 5 ). In contrast, our approach deterministically produces ( X s , E s ) (X_s,E_s) ( X s , E s ) ( X t , E t ) (X_t,E_t) ( X t , E t )
Since the ERF output at time 0 0 0 t t t
p 0 ∣ t ϕ ( x 0 ∣ x t , e t ) : = ∏ ℓ = 1 L p 0 ∣ t ϕ ( x 0 ℓ ∣ x t , e t ) . p^\phi_{0\mid t}(x_0\mid x_t,e_t):=\prod_{\ell=1}^{L}p^\phi_{0\mid t}(x^\ell_0\mid x_t,e_t). p 0 ∣ t ϕ ( x 0 ∣ x t , e t ) := ℓ = 1 ∏ L p 0 ∣ t ϕ ( x 0 ℓ ∣ x t , e t ) . For one current parameter ϕ \phi ϕ ϕ ˉ \bar{\phi} ϕ ˉ
L C D ( ϕ ; ϕ ˉ ) = E [ KL ( p 0 ∣ s ϕ ˉ ( ⋅ ∣ X s , E s ) ∥ p 0 ∣ t ϕ ( ⋅ ∣ X t , E t ) ) ] (16) \mathcal{L}_{\mathrm{CD}}(\phi;\bar{\phi})=\mathbb{E}\left[\operatorname{KL}\left(p^{\bar{\phi}}_{0\mid s}(\cdot\mid X_s,E_s)\Vert p^\phi_{0\mid t}(\cdot\mid X_t,E_t)\right)\right] \tag*{(16)} L CD ( ϕ ; ϕ ˉ ) = E [ KL ( p 0 ∣ s ϕ ˉ ( ⋅ ∣ X s , E s ) ∥ p 0 ∣ t ϕ ( ⋅ ∣ X t , E t ) ) ] (16) where ( X t , E t ) ∼ p t ⊗ Exp ( 1 ) ⊗ L × K (X_t,E_t)\sim p_t\otimes\operatorname{Exp}(1)^{\otimes L\times K} ( X t , E t ) ∼ p t ⊗ Exp ( 1 ) ⊗ L × K ( X s , E s ) = Ψ s ∣ t ϕ ( X t , E t ) (X_s,E_s)=\Psi^\phi_{s\mid t}(X_t,E_t) ( X s , E s ) = Ψ s ∣ t ϕ ( X t , E t ) t t t Unif ( δ , 1 ) \operatorname{Unif}(\delta,1) Unif ( δ , 1 ) s = t − δ s=t-\delta s = t − δ δ \delta δ ϕ ˉ \bar{\phi} ϕ ˉ [33 ] , for the minimizer to be exact, we need to impose the boundary condition p 0 ∣ 0 ϕ ( ⋅ ∣ x 0 , e 0 ) = δ x 0 p^\phi_{0\mid0}(\cdot\mid x_0,e_0)=\delta_{x_0} p 0 ∣ 0 ϕ ( ⋅ ∣ x 0 , e 0 ) = δ x 0 s = 0 s=0 s = 0 ( X t , E t ) (X_t,E_t) ( X t , E t ) t t t p 0 p_0 p 0 ( X s , E s ) (X_s, E_s) ( X s , E s ) ( X t , E t ) (X_t, E_t) ( X t , E t ) (32) before evaluating the loss. This is because the teacher was trained to model a leave-one-out denoiser while the quantity we want the student to learn is closer to a denoiser type prediction so this conversion removes a slight mismatch at least towards the start of training. The fixed-teacher and joint-refresh procedures are given in Algorithms 4 and 5.
Sampling. We consider the refresh and bridge transitions for the discrete state which are respectively ∑ x 0 q s ∣ 0 ( x s ∣ x 0 ) p 0 ∣ t ϕ ( x 0 ∣ x t , e t ) \sum_{x_0}q_{s|0}(x_s|x_0)p^\phi_{0|t}(x_0|x_t,e_t) ∑ x 0 q s ∣0 ( x s ∣ x 0 ) p 0∣ t ϕ ( x 0 ∣ x t , e t ) ∑ x 0 q s ∣ 0 , t ( x s ∣ x 0 , x t ) p 0 ∣ t ϕ ( x 0 ∣ x t , e t ) \sum_{x_0}q_{s|0,t}(x_s|x_0,x_t)p^\phi_{0|t}(x_0|x_t,e_t) ∑ x 0 q s ∣0 , t ( x s ∣ x 0 , x t ) p 0∣ t ϕ ( x 0 ∣ x t , e t )
Related Work
CTMC sampling and discretization. Campbell et al. [3 ] formulated discrete diffusion in continuous time, connecting their sampling to exact CTMC simulation (Gillespie, [14 ] ) and τ \tau τ [15 ] ). The modified next-reaction method (Anderson, [1 ] ) preserves integrated-rate clocks across jumps but resamples the firing channel’s exponential threshold. ERF instead becomes deterministic after a single random initialization, recycling residuals while preserving the exact one-time CTMC marginals, though generally with a different trajectory law. In practice, we freeze only the leave-one-out denoiser over each step while retaining the known time and state dependence of the rates, extending Euler/τ \tau τ
Deterministic sampling of discrete diffusion models. Zhang et al. (2023) introduce net-flux rates (15) to reduce jumps in a discrete analogue of the probability-flow ODE; see Appendix § 13 , but the resulting process remains stochastic. Eyring et al. [9 ] obtain deterministic trajectories by reusing Gumbel noise across discretization steps, potentially drifting from the target distribution. ERF instead gives deterministic transport after initialization while exactly preserving one-time marginals.
Distillation of discrete diffusion models. Distillation reduces sampling steps through distribution- or trajectory-based objectives. D-MMD (Hoogeboom et al., [20 ] ) uses discrete moment matching, while IDLM (Li et al., [21 ] ) uses a path space KL objective related to distribution matching (Yin et al., 2023). Both train auxiliary denoisers on student samples. Trajectory-based methods build on progressive and consistency distillation (Salimans & Ho, [28 ] ; Song et al., [33 ] ): SDTT (Deschenaux & Gulcehre, [7 ] ) aggregates teacher probabilities across stochastic unmasking steps, while DCD (Sahoo et al., [27 ] ) matches predictions along projected Gaussian trajectories with shared clean states and noise. These trajectories remain stochastic, leading to heuristic or biased objectives. Conditioning on exponential residuals instead makes the teacher trajectory deterministic given ( X t , E t ) (X_t, E_t) ( X t , E t )
Experiments
Our experiments first study the empirical consequences of ERF’s new joint law, showing that the resulting sampler achieves generation quality comparable to standard ancestral samplers at matched NFE. We then show that ERF consistency distillation (ERC) enables competitive generation with fewer NFEs than the teacher model.
General setting. We consider masked and uniform diffusion models trained on OpenWebText (OWT; Gokaslan & Cohen, [16 ] ) and TinyGSM (Liu et al., [22 ] ), using DDiT backbones with 170M and 168M parameters, respectively. OWT generation is unconditional with sequences of length 1,024; we measure GPT-2 Large generative perplexity (Radford et al., [25 ] ) together with 1-gram entropy. TinyGSM models generate Python solutions on a 512-token canvas with fixed question tokens, and we evaluate answer accuracy on GSM8K (Cobbe et al., [5 ] ), using all 1,319 test questions for the sampling comparison and the same first 1,024 questions for the main distillation comparison. For sampling, we also evaluate the released instruction-tuned DiffusionGemma model, with 26B total and approximately 4B active parameters ([17 ] ), on MMLU-Pro ((Wang et al., 2024)).
Evaluation of the ERF sampler
Sampling quality. Within each diffusion family, we compare ERF and ancestral sampling using the same pretrained checkpoint at matched NFE budgets. We use a uniform time grid and compute the ERF Uniform jump times in line 3 of Algorithm 1 using 40 bisection iterations per solve. Further evaluations appear in Appendix § 16 .§ 1 . The results are given in Figure 2 . ERF achieves sampling quality comparable to the ancestral MDM and UDM samplers across the evaluated datasets and NFE budgets. On TinyGSM, the samplers exhibit similar accuracy and temperature dependence; on OWT, their entropy–perplexity curves across temperatures largely overlap within each diffusion family.
Figure 2. Sampling quality at matched NFE on TinyGSM (top) and OpenWebText (bottom). Dashed curves denote ancestral samplers; solid curves denote ERF. TinyGSM bands are 95% bootstrap intervals over questions. OWT curves trace the generative perplexity versus 1-gram entropy frontier by sweeping the temperature in [0.8, 1.1]; ⋆ refers to temperature 1, and the vertical dash-dotted line marks the OWT dataset entropy.
The DiffusionGemma results support the same conclusion at a larger model scale (Table 1 ). At 48 NFEs and temperature 1, ERF Uniform and U-NetFlux are within 0.7 percentage points of the UDM ancestral sampler on MMLU-Pro questions. See Appendix § 17 for experiment details.
Table 1. DiffusionGemma on 140 MMLU-Pro questions and three paired seeds. Brackets give 95% question-cluster bootstrap intervals.
ERF Uniform vs U-NetFlux. The U-NetFlux version is the more efficient ERF sampler for uniform diffusion because we can solve explicitly for the crossing time in line 3 without bisection; see Appendix § 13 . Furthermore, it makes substantially fewer jumps and thus requires fewer solves. At 8 NFEs, trajectories average approximately one jump per generated position, compared with seven for Uniform, yielding speedups of 2.83 × 2.83\times 2.83 × 2.00 × 2.00\times 2.00 × Table 2 ). This gain comes while maintaining similar sampling quality in Figure 2 . We also evaluate ERF Uniform with a cap on jumps within the interval [ s i , t i ] [s_i, t_i] [ s i , t i ] Figure 10 and Figure 11 show its effect on sampling quality.
Table 2. Runtime and jump averages at 8 NFEs over three batches; times are seconds per sequence and jumps are per generated position.
Figure 10. The full TinyGSM substep comparison. Curves show GSM8K accuracy at fixed NFE as the sampling temperature varies; bands are pointwise 95% bootstrap intervals. Each legend entry specifies the number of substeps per frozen-model interval. Missing high-substep settings at larger NFE budgets are left absent.
Figure 11. The full OpenWebText substep comparison. Curves show the entropy–perplexity trade-off over eight temperatures, with stars at 1. The inverted logarithmic perplexity axis retains the observed range in each panel. Only completed evaluations are drawn, with lines interrupted at missing temperatures.
Distillation
Setting and baselines. We compare ERC with SDTT [7 ] for masked diffusion, DCD [27 ] for uniform diffusion, and D-MMD [20 ] and iDLM [21 ] for both families. All results presented are from our own reimplementations and training runs, with all the students initialized from the corresponding pretrained teacher. Within each dataset and diffusion family, the methods share the same teacher checkpoint. For both MDM and UDM, the teacher models are trained using the standard ELBOs; Sahoo et al. [26 ] , Equation (11) and Schiff et al. [29 ] , Equation (19) . In particular, for UDM our pre-trained model is a LOO denoiser. All the implementation details are given in Appendices § 14 .§ 5 and § 14 .§ 6 .
Training. ERC conditions the student on Gumbel variables − log E t -\log E_t − log E t δ ∈ { 1 / 8 , 1 / 16 , 1 / 32 , 1 / 64 } \delta\in\{1/8, 1/16, 1/32, 1/64\} δ ∈ { 1/8 , 1/16 , 1/32 , 1/64 } (32) . This gave improved results on all our experiments. Appendix § 16 .§ 5 gives the training and evaluation details. We disable the top-p p p [20 ] and fix the shared teacher’s temperature to 1, so that all methods distill the same teacher distribution.
Sampling protocol. We evaluate all methods and the teacher with both refresh and bridge sampling at matched NFE, sweeping a common temperature grid to account for differences in logit sharpness across objectives. On TinyGSM, we select one sampler, temperature placement and temperature per method, together with one ERC gap δ \delta δ [24 ] . In the TinyGSM legend, B and R denote bridge and refresh sampling; the number gives the selected inference temperature. The OWT legend marks every method with B to indicate bridge sampling.
Results. On TinyGSM, UDM ERC doubles the accuracy of the strongest competing baseline at 2 and 4 NFEs and also records higher accuracy at 8 and 16; by 32 NFEs its accuracy is close to the teacher’s. In this evaluation, it is the only method with accuracy above the teacher at some NFE budgets when fully sweeping temperature and samplers (even for the teacher). For MDM, ERC approximately matches the second-best baseline, see Figure 3 . On OWT we see the same trend, UDM ERC has lower perplexity at matched entropy than the other baselines on the full frontier, while MDM ERC approximately matches the second-best baseline, see Figure 4 . On Sudoku, ERC gives modest but consistent gains over the other methods in many evaluated settings. The complete sampler comparisons and Sudoku results appear in Appendices § 16 .§ 6 and § 16 .9. Across these comparisons, the largest ERC gains occur with uniform diffusion, while masked-diffusion results remain competitive with the considered baselines.
Figure 3. TinyGSM distillation at 2–64 NFEs. Each curve fixes its sampler and temperature to maximize mean accuracy across NFEs on the same 1,024 questions. ERC selects δ = 1 / 64 \delta= 1/64 δ = 1/64 1 / 8 1/8 1/8 R R R
Figure 4. OWT distillation with bridge sampling at 4, 8, 16, and 64 NFEs, with students trained for 100 GPU-hours. Each point uses 2,048 samples, with temperature applied to denoiser probabilities over [0.8, 1.2]; ⋆ marks temperature 1, and the vertical dash-dotted line marks the OWT dataset entropy. B refers to bridge sampling.
Ablations. We vary how the target and ERF transport (providing X s X_s X s X t X_t X t (16) ) are updated, the training gap δ \delta δ § 16 .§ 7 to § 16 .10.
Conclusion
In this work, we show that the prescribed marginals of a discrete diffusion can be transported by deterministic dynamics once the token state is augmented with continuous residuals. This construction generalizes Gillespie’s algorithm by recycling the exponential variables drawn at every jump according to an appropriate ODE. This gives a pathwise target for consistency distillation. In practice, leave-one-out freezing retains the analytically known time dependence of the rates and is exact for factorized data. Across the evaluated tasks, ERF achieves sampling quality comparable to ancestral updates at matched model evaluations, and its consistency objective yields its clearest few-step gains for uniform diffusion.
AI USE STATEMENT
A generative AI assistant was used to support literature searches, discuss mathematical expressions and experimental results, and assist with manuscript drafting, editing, and LaTeX preparation. The authors take responsibility for all claims, citations, derivations, reported results, and the final content of the paper.
ACKNOWLEDGMENTS
We thank Patrick Pynadath and Jawad Chemaou for their useful comments on the presentation of this paper.
AD received funding from the Fondation de l’École polytechnique through its Servir la science campaign. AD is supported by the France 2030 program under reference ANR-25-PEIA-0001 (THEOREM project). AD is funded by the European Union (ERC-2022-SYG-OCEAN-101071601). AD is supported by Hi! Paris and the Agence Nationale de la Recherche (Grant 11-LABX-0047). This work received government funding administered by the National Research Agency (ANR) under the France 2030 program Hi! PARIS , grant number ANR-23-IACL-0005.
Rates and jump times used in the discretization
Fix [ s , t ] = [ s i , t i ] [s,t]=[s_i,t_i] [ s , t ] = [ s i , t i ] X t i = x X_{t_i}=x X t i = x x ^ 0 θ ( x , t i ) \hat{x}_0^\theta(x,t_i) x ^ 0 θ ( x , t i ) z z z τ ∈ [ s , t ] \tau\in[s,t] τ ∈ [ s , t ] u ∈ [ s , τ ] u \in[s,\tau] u ∈ [ s , τ ] E r ℓ j E_r^{\ell j} E r ℓ j u u u τ \tau τ E r ℓ j E_{r}^{\ell j} E r ℓ j § 12 .§ 5 and § 13 .
For MDM, only z ℓ = m z^{\ell}=m z ℓ = m j ≠ m j\ne m j = m
∫ u r Q ˉ v θ , ℓ ( j ; t i ∣ z ) d v = x ^ 0 θ ( x , t i ) ℓ j log ( 1 − α r 1 − α u ) , ( z ℓ = m ) , \int_{u}^{r}\bar{Q}_{v}^{\theta,\ell}(j;t_i\mid z)\,\mathrm{d}v=\hat{x}_{0}^{\theta}(x,t_i)^{\ell j}\log\left(\frac{1-\alpha_r}{1-\alpha_u}\right),\qquad(z^{\ell}=m), ∫ u r Q ˉ v θ , ℓ ( j ; t i ∣ z ) d v = x ^ 0 θ ( x , t i ) ℓ j log ( 1 − α u 1 − α r ) , ( z ℓ = m ) , ∫ u r λ v ℓ ( z ℓ ) d v = 1 z ℓ ≠ m log ( α u α r ) . \int_{u}^{r}\lambda_{v}^{\ell}(z^{\ell})\,\mathrm{d}v=\mathbf{1}_{z^{\ell}\ne m}\log\left(\frac{\alpha_u}{\alpha_r}\right). ∫ u r λ v ℓ ( z ℓ ) d v = 1 z ℓ = m log ( α r α u ) . If x ^ 0 θ ( x , t i ) ℓ j > 0 \hat{x}_{0}^{\theta}(x,t_i)^{\ell j}>0 x ^ 0 θ ( x , t i ) ℓ j > 0 E r ℓ j ≤ x ^ 0 θ ( x , t i ) ℓ j log ( ( 1 − α r ) / ( 1 − α s ) ) E_{r}^{\ell j}\le\hat{x}_{0}^{\theta}(x,t_i)^{\ell j}\log((1-\alpha_r)/(1-\alpha_s)) E r ℓ j ≤ x ^ 0 θ ( x , t i ) ℓ j log (( 1 − α r ) / ( 1 − α s ))
T ℓ j = α − 1 ( 1 − ( 1 − α r ) exp { − E r ℓ j x ^ 0 θ ( x , t i ) ℓ j } ) . T^{\ell j}=\alpha^{-1}\left(1-(1-\alpha_r)\exp\left\{-\frac{E_{r}^{\ell j}}{\hat{x}_{0}^{\theta}(x,t_i)^{\ell j}}\right\}\right). T ℓ j = α − 1 ( 1 − ( 1 − α r ) exp { − x ^ 0 θ ( x , t i ) ℓ j E r ℓ j } ) . and T ℓ j = − ∞ T^{\ell j}=-\infty T ℓ j = − ∞
For UDM with net flux and j ≠ z ℓ j\ne z^{\ell} j = z ℓ
∫ u r Q ˉ v N F , θ , ℓ ( j ; t i ∣ z ) d v = [ x ^ 0 θ ( x , t i ) ℓ j − x ^ 0 θ ( x , t i ) ℓ z ℓ ] + K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 log ( 1 + ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) α u 1 + ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) α r ) , \int_{u}^{r}\bar{Q}_{v}^{\mathrm{NF},\theta,\ell}(j;t_i\mid z)\,\mathrm{d}v=
\frac{\left[\hat{x}_{0}^{\theta}(x,t_i)^{\ell j}-\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}\right]_{+}}{K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}-1}
\log\left(\frac{1+\left(K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}-1\right)\alpha_u}{1+\left(K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}-1\right)\alpha_r}\right), ∫ u r Q ˉ v NF , θ , ℓ ( j ; t i ∣ z ) d v = K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 [ x ^ 0 θ ( x , t i ) ℓ j − x ^ 0 θ ( x , t i ) ℓ z ℓ ] + log 1 + ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) α r 1 + ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) α u , ∫ u r b v N F , ℓ ( z ) d v = ∑ j ∈ V ∖ { z ℓ } [ x ^ 0 θ ( x , t i ) ℓ z ℓ − x ^ 0 θ ( x , t i ) ℓ j ] + K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 log ( 1 + ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) α u 1 + ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) α r ) . \int_{u}^{r}b_{v}^{\mathrm{NF},\ell}(z)\,\mathrm{d}v=
\frac{\sum_{j\in\mathcal{V}\setminus\{z^{\ell}\}}\left[\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}-\hat{x}_{0}^{\theta}(x,t_i)^{\ell j}\right]_{+}}{K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}-1}
\log\left(\frac{1+\left(K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}-1\right)\alpha_u}{1+\left(K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}-1\right)\alpha_r}\right). ∫ u r b v NF , ℓ ( z ) d v = K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ∑ j ∈ V ∖ { z ℓ } [ x ^ 0 θ ( x , t i ) ℓ z ℓ − x ^ 0 θ ( x , t i ) ℓ j ] + log 1 + ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) α r 1 + ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) α u . Here b N F , ℓ ( z ) b^{\mathrm{NF},\ell}(z) b NF , ℓ ( z ) ℓ \ell ℓ K x ^ 0 θ ( x , t i ) ℓ z ℓ = 1 K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}=1 K x ^ 0 θ ( x , t i ) ℓ z ℓ = 1 K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}-1 K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 α u − α r \alpha_u-\alpha_r α u − α r x ^ 0 θ ( x , t i ) ℓ j > x ^ 0 θ ( x , t i ) ℓ z ℓ \hat{x}_{0}^{\theta}(x,t_i)^{\ell j}>\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}} x ^ 0 θ ( x , t i ) ℓ j > x ^ 0 θ ( x , t i ) ℓ z ℓ u = s u=s u = s E r ℓ j E_{r}^{\ell j} E r ℓ j
T ℓ j = α − 1 ( ( 1 + ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) α r ) exp { ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) E r ℓ j x ^ 0 θ ( x , t i ) ℓ j − x ^ 0 θ ( x , t i ) ℓ z ℓ } − 1 K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) . T^{\ell j}=\alpha^{-1}\left(\frac{\left(1+\left(K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}-1\right)\alpha_r\right)\exp\left\{\frac{\left(K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}-1\right)E_{r}^{\ell j}}{\hat{x}_{0}^{\theta}(x,t_i)^{\ell j}-\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}}\right\}-1}{K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}-1}\right). T ℓ j = α − 1 K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ( 1 + ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) α r ) exp ⎩ ⎨ ⎧ x ^ 0 θ ( x , t i ) ℓ j − x ^ 0 θ ( x , t i ) ℓ z ℓ ( K x ^ 0 θ ( x , t i ) ℓ z ℓ − 1 ) E r ℓ j ⎭ ⎬ ⎫ − 1 . For K x ^ 0 θ ( x , t i ) ℓ z ℓ = 1 K\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}=1 K x ^ 0 θ ( x , t i ) ℓ z ℓ = 1 T ℓ j = α − 1 ( α r + E r ℓ j x ^ 0 θ ( x , t i ) ℓ j − x ^ 0 θ ( x , t i ) ℓ z ℓ ) T^{\ell j}=\alpha^{-1}\left(\alpha_r+\frac{E_{r}^{\ell j}}{\hat{x}_{0}^{\theta}(x,t_i)^{\ell j}-\hat{x}_{0}^{\theta}(x,t_i)^{\ell z^{\ell}}}\right) T ℓ j = α − 1 ( α r + x ^ 0 θ ( x , t i ) ℓ j − x ^ 0 θ ( x , t i ) ℓ z ℓ E r ℓ j ) T ℓ j = − ∞ T^{\ell j}=-\infty T ℓ j = − ∞ z z z
Parallel ERF sampling
Algorithm 2 applies the leave-one-out-frozen rates above independently at each position on each grid interval. In the algorithm, Q ˉ v ℓ ( j ; t i ∣ X s i ) \bar{Q}_{v}^{\ell}(j;t_i\mid X_{s_i}) Q ˉ v ℓ ( j ; t i ∣ X s i ) b v ℓ ( X s i ) b_{v}^{\ell}(X_{s_i}) b v ℓ ( X s i ) λ r ℓ ( X s i ) \lambda_{r}^{\ell}(X_{s_i}) λ r ℓ ( X s i ) b r N F , ℓ ( X s i ) b_{r}^{\mathrm{NF},\ell}(X_{s_i}) b r NF , ℓ ( X s i ) − ∞ -\infty − ∞
Bridge and simplex extension
For 0 ≤ s < t ≤ 1 0\le s<t\le1 0 ≤ s < t ≤ 1 α s > 0 \alpha_s>0 α s > 0
q s ∣ 0 , t ( x s ∣ x 0 , x t ) = ∏ ℓ = 1 L q s ∣ 0 , t ℓ ( x s ℓ ∣ x 0 ℓ , x t ℓ ) . q_{s\mid0,t}(x_s\mid x_0,x_t)=\prod_{\ell=1}^{L}q_{s\mid0,t}^{\ell}(x_s^{\ell}\mid x_0^{\ell},x_t^{\ell}). q s ∣ 0 , t ( x s ∣ x 0 , x t ) = ℓ = 1 ∏ L q s ∣ 0 , t ℓ ( x s ℓ ∣ x 0 ℓ , x t ℓ ) . For x 0 , x t ∈ X x_0,x_t\in\mathcal{X} x 0 , x t ∈ X ℓ \ell ℓ Algorithm 2 Parallel ERF sampling with leave-one-out-frozen rates
Require: Grid 0 = t 0 < ⋯ < t n = 1 0=t_0<\cdots<t_n=1 0 = t 0 < ⋯ < t n = 1
Draw X 1 ∼ p 1 X_1\sim p_1 X 1 ∼ p 1 E T 0 ⋆ ℓ j ∼ i . i . d . Exp ( 1 ) E_{T_0^\star}^{\ell j}\mathrel{\overset{\mathrm{i.i.d.}}{\sim}}\operatorname{Exp}(1) E T 0 ⋆ ℓ j ∼ i.i.d. Exp ( 1 ) ( ℓ , j ) (\ell,j) ( ℓ , j )
for i = n , … , 1 i=n,\ldots,1 i = n , … , 1 do
Cache x ~ 0 ℓ ( X t i , t i ) \widetilde{x}_0^\ell(X_{t_i},t_i) x 0 ℓ ( X t i , t i ) [ s i , t i ] [s_i,t_i] [ s i , t i ]
( X s i , E s i ) ← ( X t i , E t i ) (X_{s_i},E_{s_i})\leftarrow(X_{t_i},E_{t_i}) ( X s i , E s i ) ← ( X t i , E t i )
for all ℓ ∈ ⟦ 1 , L ⟧ \ell\in\llbracket1,L\rrbracket ℓ ∈ [ [ 1 , L ] ] in parallel do
τ ⋆ ← t i \tau^\star\leftarrow t_i τ ⋆ ← t i
while τ ⋆ > s i \tau^\star>s_i τ ⋆ > s i do
τ ℓ j ← sup { u ∈ [ s i , τ ⋆ ] : ∫ u τ ⋆ Q ‾ r ℓ ( j ; t i ∣ X s i ) d r ≥ E s i ℓ j } \tau^{\ell j}\leftarrow\sup\left\{u\in[s_i,\tau^\star]:\int_u^{\tau^\star}\overline{Q}^{\ell}_{r}(j;t_i\mid X_{s_i})\,\mathrm{d}r\ge E_{s_i}^{\ell j}\right\} τ ℓ j ← sup { u ∈ [ s i , τ ⋆ ] : ∫ u τ ⋆ Q r ℓ ( j ; t i ∣ X s i ) d r ≥ E s i ℓ j } j ≠ X s i ℓ j\ne X_{s_i}^{\ell} j = X s i ℓ
j ⋆ ← argmax j ≠ X s i ℓ τ ℓ j j^\star\leftarrow\underset{j\ne X_{s_i}^{\ell}}{\operatorname{argmax}}\ \tau^{\ell j} j ⋆ ← j = X s i ℓ argmax τ ℓ j
τ n e w ⋆ ← max { s i , τ ℓ j ⋆ } \tau_{\mathrm{new}}^\star\leftarrow\max\{s_i,\tau^{\ell j^\star}\} τ new ⋆ ← max { s i , τ ℓ j ⋆ }
E s i ℓ j ← E s i ℓ j − ∫ τ n e w ⋆ τ ⋆ Q ‾ r ℓ ( j ; t i ∣ X s i ) d r E_{s_i}^{\ell j}\leftarrow E_{s_i}^{\ell j}-\int_{\tau_{\mathrm{new}}^\star}^{\tau^\star}\overline{Q}^{\ell}_{r}(j;t_i\mid X_{s_i})\,\mathrm{d}r E s i ℓ j ← E s i ℓ j − ∫ τ new ⋆ τ ⋆ Q r ℓ ( j ; t i ∣ X s i ) d r j ≠ X s i ℓ j\ne X_{s_i}^{\ell} j = X s i ℓ
E s i ℓ j ← E s i ℓ j + ∫ τ n e w ⋆ τ ⋆ b r ℓ ( X s i ) d r E_{s_i}^{\ell j}\leftarrow E_{s_i}^{\ell j}+\int_{\tau_{\mathrm{new}}^\star}^{\tau^\star}b_r^\ell(X_{s_i})\,\mathrm{d}r E s i ℓ j ← E s i ℓ j + ∫ τ new ⋆ τ ⋆ b r ℓ ( X s i ) d r j = X s i ℓ j=X_{s_i}^{\ell} j = X s i ℓ
if τ ℓ j ⋆ ≥ s i \tau^{\ell j^\star}\ge s_i τ ℓ j ⋆ ≥ s i then
X s i ℓ ← j ⋆ X_{s_i}^{\ell}\leftarrow j^\star X s i ℓ ← j ⋆
end if
τ ⋆ ← τ n e w ⋆ \tau^\star\leftarrow\tau_{\mathrm{new}}^\star τ ⋆ ← τ new ⋆
end while
end for
end for
return X 0 , E 0 X_0,E_0 X 0 , E 0
0 0 0
q s ∣ 0 , t ℓ ( x 0 ℓ ∣ x t 0 , x t ℓ ) = Cat ( ⋅ ; [ α t ∣ s x ~ t 0 + ( 1 − α t ∣ s ) ⟨ x ~ t 0 , π ℓ ⟩ 1 K ] ⊙ [ α s x ~ 0 0 + ( 1 − α s ) π ℓ ] α t ⟨ x ~ t 0 , x ~ 0 0 ⟩ + ( 1 − α t ) ⟨ x ~ t 0 , π ℓ ⟩ ) , (17) q_{s\mid0,t}^{\ell}(x_0^\ell\mid x_t^0,x_t^\ell)=\operatorname{Cat}\left(\mathord{\cdot};\frac{\left[\alpha_{t\mid s}\widetilde{x}_t^0+(1-\alpha_{t\mid s})\langle\widetilde{x}_t^0,\pi^\ell\rangle\mathbf{1}_K\right]\odot\left[\alpha_s\widetilde{x}_0^0+(1-\alpha_s)\pi^\ell\right]}{\alpha_t\langle\widetilde{x}_t^0,\widetilde{x}_0^0\rangle+(1-\alpha_t)\langle\widetilde{x}_t^0,\pi^\ell\rangle}\right), \tag*{(17)} q s ∣ 0 , t ℓ ( x 0 ℓ ∣ x t 0 , x t ℓ ) = Cat ( ⋅ ; α t ⟨ x t 0 , x 0 0 ⟩ + ( 1 − α t ) ⟨ x t 0 , π ℓ ⟩ [ α t ∣ s x t 0 + ( 1 − α t ∣ s ) ⟨ x t 0 , π ℓ ⟩ 1 K ] ⊙ [ α s x 0 0 + ( 1 − α s ) π ℓ ] ) , (17) where x ~ 0 0 = x ~ 0 \widetilde{x}_0^0=\widetilde{x}^0 x 0 0 = x 0 x ~ t ℓ = x ~ ℓ \widetilde{x}_t^\ell=\widetilde{x}^\ell x t ℓ = x ℓ ν ∈ Δ K \nu\in\Delta_K ν ∈ Δ K x 0 ℓ x_0^\ell x 0 ℓ ν \nu ν q s ∣ 0 , t ( x s ℓ ∣ ν , x t ℓ ) = q t ∣ s ( x t ℓ ∣ x s ℓ ) q s ∣ 0 ( x s ℓ ∣ ν ) / q t ∣ 0 ( x t ℓ ∣ ν ) q_{s\mid0,t}(x_s^\ell\mid\nu,x_t^\ell)=q_{t\mid s}(x_t^\ell\mid x_s^\ell)q_{s\mid0}(x_s^\ell\mid\nu)/q_{t\mid0}(x_t^\ell\mid\nu) q s ∣ 0 , t ( x s ℓ ∣ ν , x t ℓ ) = q t ∣ s ( x t ℓ ∣ x s ℓ ) q s ∣ 0 ( x s ℓ ∣ ν ) / q t ∣ 0 ( x t ℓ ∣ ν ) q r ∣ 0 ( ⋅ ∣ ν ) = Cat ( ⋅ ; α r ν + ( 1 − α r ) π ℓ ) q_{r\mid0}(\cdot\mid\nu)=\operatorname{Cat}(\cdot;\alpha_r\nu+(1-\alpha_r)\pi^\ell) q r ∣ 0 ( ⋅ ∣ ν ) = Cat ( ⋅ ; α r ν + ( 1 − α r ) π ℓ ) (17) has nonnegative entries whose sum is the denominator, since α t ∣ s α s = α t \alpha_{t\mid s}\alpha_s=\alpha_t α t ∣ s α s = α t X \mathcal{X} X Cat ( ⋅ ; x ~ t ℓ ) \operatorname{Cat}(\mathord{\cdot};\widetilde{x}_t^\ell) Cat ( ⋅ ; x t ℓ ) [18 ] , define x ~ 0 ∣ t ℓ loo ( x t ℓ ) : = E [ X ~ 0 ℓ ∣ X t − ℓ = x t − ℓ ] \widetilde{x}_{0\mid t}^{\ell\operatorname{loo}}(x_t^\ell):=\mathbb{E}[\widetilde{X}_0^\ell\mid X_t^{-\ell}=x_t^{-\ell}] x 0 ∣ t ℓ loo ( x t ℓ ) := E [ X 0 ℓ ∣ X t − ℓ = x t − ℓ ] X t − ℓ X_t^{-\ell} X t − ℓ X t ℓ X_t^\ell X t ℓ p t ( x t ) > 0 p_t(x_t)>0 p t ( x t ) > 0 p s ∣ t ℓ ( x s ℓ ∣ x t ) = q s ∣ 0 , t ℓ ( x s ℓ ∣ x ~ 0 ∣ t ℓ loo ( x t ℓ ) , x t ℓ ) p_{s\mid t}^{\ell}(x_s^\ell\mid x_t)=q_{s\mid0,t}^{\ell}(x_s^\ell\mid\widetilde{x}_{0\mid t}^{\ell\operatorname{loo}}(x_t^\ell),x_t^\ell) p s ∣ t ℓ ( x s ℓ ∣ x t ) = q s ∣ 0 , t ℓ ( x s ℓ ∣ x 0 ∣ t ℓ loo ( x t ℓ ) , x t ℓ ) (3) .
CTMC integration schemes
The Gillespie’s first reaction form (Gillespie, [15 ] ), can be used to simulate ( X u ) u ∈ [ s , t ] (X_u)_{u\in[s,t]} ( X u ) u ∈ [ s , t ] X t ∼ p t X_t\sim p_t X t ∼ p t ( Q ‾ u ) u ∈ [ s , t ] (\overline{Q}_u)_{u\in[s,t]} ( Q u ) u ∈ [ s , t ] ( T k ⋆ ) k ≤ N (T_k^\star)_{k\le N} ( T k ⋆ ) k ≤ N ( X T k ⋆ ) k ≤ N (X_{T_k^\star})_{k\le N} ( X T k ⋆ ) k ≤ N N ∞ N_\infty N ∞
Initialization. Set T 0 ⋆ = t T_0^\star=t T 0 ⋆ = t N ∞ = ∞ N_\infty=\infty N ∞ = ∞ X T 0 ⋆ ∼ p t X_{T_0^\star}\sim p_t X T 0 ⋆ ∼ p t ( ℓ , j ) ∈ ⟦ 1 , L ⟧ × ⟦ 1 , K ⟧ (\ell,j)\in\llbracket1,L\rrbracket\times\llbracket1,K\rrbracket ( ℓ , j ) ∈ [ [ 1 , L ] ] × [ [ 1 , K ] ] E T 0 ⋆ ℓ j ∼ Exp ( 1 ) E_{T_0^\star}^{\ell j}\sim\operatorname{Exp}(1) E T 0 ⋆ ℓ j ∼ Exp ( 1 )
Induction. At stage k ≥ 0 k \ge0 k ≥ 0 T k ⋆ T_k^\star T k ⋆ X T k ⋆ X_{T_k^\star} X T k ⋆ E T k ⋆ ℓ j E_{T_k^\star}^{\ell j} E T k ⋆ ℓ j ( ℓ , j ) ∈ ⟦ 1 , L ⟧ × ⟦ 1 , K ⟧ (\ell,j) \in\llbracket1,L\rrbracket\times\llbracket1,K\rrbracket ( ℓ , j ) ∈ [ [ 1 , L ] ] × [ [ 1 , K ] ]
T k + 1 ℓ j = sup { u ∈ [ s , T k ⋆ ] : ∫ u T k ⋆ Q ~ r ℓ ( j ∣ X T k ⋆ ) d r ≥ E T k ⋆ ℓ j } , ( ℓ k + 1 ⋆ , j k + 1 ⋆ ) ∈ argmax ( ℓ , j ) T k + 1 ℓ j . (18) T_{k+1}^{\ell j}=\sup\left\{u\in[s,T_k^\star]:\int_u^{T_k^\star}\widetilde{Q}_r^\ell(j\mid X_{T_k^\star})\,\mathrm{d}r\ge E_{T_k^\star}^{\ell j}\right\},\qquad(\ell_{k+1}^\star,j_{k+1}^\star)\in\underset{(\ell,j)}{\operatorname{argmax}}\,T_{k+1}^{\ell j}. \tag*{(18)} T k + 1 ℓ j = sup { u ∈ [ s , T k ⋆ ] : ∫ u T k ⋆ Q r ℓ ( j ∣ X T k ⋆ ) d r ≥ E T k ⋆ ℓ j } , ( ℓ k + 1 ⋆ , j k + 1 ⋆ ) ∈ ( ℓ , j ) argmax T k + 1 ℓ j . (18) We use here the convention sup ∅ = − ∞ \sup\varnothing=-\infty sup ∅ = − ∞
If T k + 1 ℓ k + 1 ⋆ j k + 1 ⋆ = − ∞ T_{k+1}^{\ell_{k+1}^\star j_{k+1}^\star}=-\infty T k + 1 ℓ k + 1 ⋆ j k + 1 ⋆ = − ∞ N ∞ = k N_\infty=k N ∞ = k X s = X T k ⋆ X_s=X_{T_k^\star} X s = X T k ⋆
Otherwise, the ( k + 1 ) (k+1) ( k + 1 ) T k + 1 ⋆ = T k + 1 ℓ k + 1 ⋆ j k + 1 ⋆ T_{k+1}^\star=T_{k+1}^{\ell_{k+1}^\star j_{k+1}^\star} T k + 1 ⋆ = T k + 1 ℓ k + 1 ⋆ j k + 1 ⋆ X T k + 1 ⋆ ℓ k + 1 ⋆ = j k + 1 ⋆ X_{T_{k+1}^\star}^{\ell_{k+1}^\star}=j_{k+1}^\star X T k + 1 ⋆ ℓ k + 1 ⋆ = j k + 1 ⋆ ( ℓ , j ) (\ell,j) ( ℓ , j ) X T k ⋆ ℓ ≠ j X_{T_k^\star}^\ell\ne j X T k ⋆ ℓ = j
E T k + 1 ⋆ ℓ j = E T k ⋆ ℓ j − ∫ T k + 1 ⋆ T k ⋆ Q ~ r ℓ ( j ∣ X T k ⋆ ) d r ≥ 0. E_{T_{k+1}^\star}^{\ell j}=E_{T_k^\star}^{\ell j}-\int_{T_{k+1}^\star}^{T_k^\star}\widetilde{Q}_r^\ell(j\mid X_{T_k^\star})\,\mathrm{d}r\ge0. E T k + 1 ⋆ ℓ j = E T k ⋆ ℓ j − ∫ T k + 1 ⋆ T k ⋆ Q r ℓ ( j ∣ X T k ⋆ ) d r ≥ 0. Finally, draw E T k + 1 ⋆ ℓ k + 1 ⋆ j k + 1 ⋆ ∼ Exp ( 1 ) E_{T_{k+1}^\star}^{\ell_{k+1}^\star j_{k+1}^\star}\sim\operatorname{Exp}(1) E T k + 1 ⋆ ℓ k + 1 ⋆ j k + 1 ⋆ ∼ Exp ( 1 )
The recursion continues until no candidate event occurs before s s s [10 ] ), N N N ( X u ) u ∈ [ s , t ] (X_u)_{u\in[s,t]} ( X u ) u ∈ [ s , t ] X t = x X_t=x X t = x
The ERF sampler
The ERF augments X u X_u X u E u ℓ j E_u^{\ell j} E u ℓ j ℓ \ell ℓ j j j ( X u , E u ) u ∈ [ s , t ] (X_u,E_u)_{u\in[s,t]} ( X u , E u ) u ∈ [ s , t ]
Initialization. Set T 0 ⋆ = t T_0^\star=t T 0 ⋆ = t X T 0 ⋆ ∼ p t X_{T_0^\star}\sim p_t X T 0 ⋆ ∼ p t E T 0 ⋆ ℓ j ∼ i . i . d . Exp ( 1 ) E_{T_0^\star}^{\ell j}\mathrel{\overset{\mathrm{i.i.d.}}{\sim}}\operatorname{Exp}(1) E T 0 ⋆ ℓ j ∼ i.i.d. Exp ( 1 ) ( ℓ , j ) ∈ ⟦ 1 , L ⟧ × ⟦ 1 , K ⟧ (\ell,j)\in\llbracket1,L\rrbracket\times\llbracket1,K\rrbracket ( ℓ , j ) ∈ [ [ 1 , L ] ] × [ [ 1 , K ] ] X T 0 ⋆ X_{T_0^\star} X T 0 ⋆
Induction. At stage k ≥ 0 k\ge0 k ≥ 0 ( T k ⋆ , X T k ⋆ , E T k ⋆ ) (T_k^\star,X_{T_k^\star},E_{T_k^\star}) ( T k ⋆ , X T k ⋆ , E T k ⋆ ) z = X T k ⋆ z=X_{T_k^\star} z = X T k ⋆ j ≠ z ℓ j\ne z^\ell j = z ℓ
T k + 1 ℓ j = sup { u ∈ [ s , T k ⋆ ] : ∫ u T k ⋆ Q ~ r ℓ ( j ∣ z ) d r ≥ E T k ⋆ ℓ j } , ( ℓ k + 1 ⋆ , j k + 1 ⋆ ) ∈ argmax ( ℓ , j ) : j ≠ z ℓ T k + 1 ℓ j . (19) T_{k+1}^{\ell j}=\sup\left\{u\in[s,T_k^\star]:\int_u^{T_k^\star}\widetilde{Q}_r^\ell(j\mid z)\,\mathrm{d}r\ge E_{T_k^\star}^{\ell j}\right\},\qquad(\ell_{k+1}^\star,j_{k+1}^\star)\in\underset{(\ell,j):j\ne z^\ell}{\operatorname{argmax}}\,T_{k+1}^{\ell j}. \tag*{(19)} T k + 1 ℓ j = sup { u ∈ [ s , T k ⋆ ] : ∫ u T k ⋆ Q r ℓ ( j ∣ z ) d r ≥ E T k ⋆ ℓ j } , ( ℓ k + 1 ⋆ , j k + 1 ⋆ ) ∈ ( ℓ , j ) : j = z ℓ argmax T k + 1 ℓ j . (19) Set T k + 1 ⋆ = T k + 1 ℓ k + 1 ⋆ j k + 1 ⋆ T_{k+1}^\star=T_{k+1}^{\ell_{k+1}^\star j_{k+1}^\star} T k + 1 ⋆ = T k + 1 ℓ k + 1 ⋆ j k + 1 ⋆
If T k + 1 ⋆ = − ∞ T_{k+1}^\star=-\infty T k + 1 ⋆ = − ∞ N = k N=k N = k X v = z X_v=z X v = z v ∈ [ s , T k ⋆ ] v\in[s,T_k^\star] v ∈ [ s , T k ⋆ ]
Otherwise, keep X v = z X_v=z X v = z v ∈ ( T k + 1 ⋆ , T k ⋆ ] v\in(T_{k+1}^\star,T_k^\star] v ∈ ( T k + 1 ⋆ , T k ⋆ ] v ∈ [ T k + 1 ⋆ , T k ⋆ ] v\in[T_{k+1}^\star,T_k^\star] v ∈ [ T k + 1 ⋆ , T k ⋆ ]
E v ℓ j = E T k ⋆ ℓ j + { − ∫ v T k ⋆ Q ~ r ℓ ( j ∣ z ) d r , j ≠ z ℓ , ∫ v T k ⋆ λ r ℓ ( z ℓ ) d r , j = z ℓ . (20) E_v^{\ell j}=E_{T_k^\star}^{\ell j}+
\begin{cases}
-\int_v^{T_k^\star}\widetilde{Q}_r^\ell(j\mid z)\,\mathrm{d}r, & j\ne z^\ell,\\
\int_v^{T_k^\star}\lambda_r^\ell(z^\ell)\,\mathrm{d}r, & j=z^\ell.
\end{cases} \tag*{(20)} E v ℓ j = E T k ⋆ ℓ j + { − ∫ v T k ⋆ Q r ℓ ( j ∣ z ) d r , ∫ v T k ⋆ λ r ℓ ( z ℓ ) d r , j = z ℓ , j = z ℓ . (20) Set X T k + 1 ⋆ ℓ k + 1 ⋆ = j k + 1 ⋆ X_{T_{k+1}^\star}^{\ell_{k+1}^\star}=j_{k+1}^\star X T k + 1 ⋆ ℓ k + 1 ⋆ = j k + 1 ⋆ X T k + 1 ⋆ m = z m X_{T_{k+1}^\star}^m=z^m X T k + 1 ⋆ m = z m m ≠ ℓ k + 1 ⋆ m\ne\ell_{k+1}^\star m = ℓ k + 1 ⋆
The recursion continues until no candidate event occurs before s s s
Gillespie's global form
The global form of Gillespie’s algorithm [14 ] uses one exponential clock per jump. As above, it constructs ( X u ) u ∈ [ s , t ] (X_u)_{u\in[s,t]} ( X u ) u ∈ [ s , t ] X t ∼ p t X_t\sim p_t X t ∼ p t
Initialization. Set T 0 ⋆ = t T_0^\star=t T 0 ⋆ = t X T 0 ⋆ ∼ p t X_{T_0^\star}\sim p_t X T 0 ⋆ ∼ p t
Algorithm 3 Gillespie discretization with partially frozen reverse rates
Require: Grid 0 = t 0 < ⋯ < t n = 1 0 = t_0 < \cdots< t_n = 1 0 = t 0 < ⋯ < t n = 1
Draw X 1 ∼ p 1 X_1 \sim p_1 X 1 ∼ p 1
for i = n , … , 1 i = n,\ldots,1 i = n , … , 1 do ▷ Discretize time
Set r ← t i r \leftarrow t_i r ← t i s ← s i s \leftarrow s_i s ← s i z ← X t i z \leftarrow X_{t_i} z ← X t i
Evaluate the learned component at ( t i , z ) (t_i,z) ( t i , z ) Q ˉ r θ ( ⋅ ; t i ∣ ⋅ ) \bar{Q}^{\theta}_{r}(\cdot;t_i\mid\cdot) Q ˉ r θ ( ⋅ ; t i ∣ ⋅ )
while r > s r > s r > s do
Draw E ℓ j E^{\ell j} E ℓ j ∼ Exp ( 1 ) \sim\operatorname{Exp}(1) ∼ Exp ( 1 ) ( ℓ , j ) (\ell,j) ( ℓ , j ) j ≠ z ℓ j \ne z^\ell j = z ℓ
T ℓ j ← sup { u ∈ [ s , r ] : ∫ u r Q ˉ r θ , ℓ ( j ; t i ∣ z ) d v ≥ E ℓ j } T^{\ell j} \leftarrow\sup\{u \in[s,r]: \int_u^r \bar{Q}^{\theta,\ell}_{r}(j;t_i\mid z)\,\mathrm{d}v \ge E^{\ell j}\} T ℓ j ← sup { u ∈ [ s , r ] : ∫ u r Q ˉ r θ , ℓ ( j ; t i ∣ z ) d v ≥ E ℓ j }
( ℓ ^ , j ^ ) ← argmax ( ℓ , j ) T ℓ j (\hat{\ell},\hat{j}) \leftarrow\operatorname*{argmax}_{(\ell,j)} T^{\ell j} ( ℓ ^ , j ^ ) ← argmax ( ℓ , j ) T ℓ j u ← T ℓ ^ j ^ u \leftarrow T^{\hat{\ell}\hat{j}} u ← T ℓ ^ j ^
if u < s u < s u < s then
r ← s r \leftarrow s r ← s
else
z ℓ ^ ← j ^ z^{\hat{\ell}} \leftarrow\hat{j} z ℓ ^ ← j ^ r ← u r \leftarrow u r ← u
end if
end while
X s i ← z X_{s_i} \leftarrow z X s i ← z
end for
return X 0 X_0 X 0
Induction. At stage k ≥ 0 k \ge0 k ≥ 0 ( T k ⋆ , X T k ⋆ ) (T^\star_k,X_{T^\star_k}) ( T k ⋆ , X T k ⋆ ) z = X T k ⋆ z=X_{T^\star_k} z = X T k ⋆ λ ˉ r ( z ) : = ∑ ℓ = 1 L ∑ j : j ≠ z ℓ Q ˉ r ℓ ( j ∣ z ) \bar{\lambda}_r(z):=\sum_{\ell=1}^{L}\sum_{j:j\ne z^\ell}\bar{Q}^{\ell}_{r}(j\mid z) λ ˉ r ( z ) := ∑ ℓ = 1 L ∑ j : j = z ℓ Q ˉ r ℓ ( j ∣ z ) E k ∼ Exp ( 1 ) E_k\sim\operatorname{Exp}(1) E k ∼ Exp ( 1 )
T k + 1 ⋆ = sup { u ∈ [ s , T k ⋆ ] : ∫ u T k ⋆ λ ˉ r ( z ) d r ≥ E k } . (21) T^\star_{k+1}=\sup\left\{u\in[s,T^\star_k]:\int_u^{T^\star_k}\bar{\lambda}_r(z)\,\mathrm{d}r\ge E_k\right\}. \tag*{(21)} T k + 1 ⋆ = sup { u ∈ [ s , T k ⋆ ] : ∫ u T k ⋆ λ ˉ r ( z ) d r ≥ E k } . (21) If T k + 1 ⋆ = − ∞ T^\star_{k+1}=-\infty T k + 1 ⋆ = − ∞ N = k N=k N = k X u = z X_u=z X u = z u ∈ [ s , T k ⋆ ] u\in[s,T^\star_k] u ∈ [ s , T k ⋆ ]
Otherwise, draw ( ℓ k + 1 ⋆ , j k + 1 ⋆ ) (\ell^\star_{k+1},j^\star_{k+1}) ( ℓ k + 1 ⋆ , j k + 1 ⋆ ) j ≠ z ℓ j\ne z^\ell j = z ℓ
P ( ( ℓ k + 1 ⋆ , j k + 1 ⋆ ) = ( ℓ , j ) ∣ T k + 1 ⋆ = u , X T k ⋆ = z ) = Q ˉ u ℓ ( j ∣ z ) λ ˉ u ( z ) . \mathbb{P}\left((\ell^\star_{k+1},j^\star_{k+1})=(\ell,j)\mid T^\star_{k+1}=u,X_{T^\star_k}=z\right)=\frac{\bar{Q}^{\ell}_{u}(j\mid z)}{\bar{\lambda}_{u}(z)}. P ( ( ℓ k + 1 ⋆ , j k + 1 ⋆ ) = ( ℓ , j ) ∣ T k + 1 ⋆ = u , X T k ⋆ = z ) = λ ˉ u ( z ) Q ˉ u ℓ ( j ∣ z ) . Set X v = z X_v=z X v = z v ∈ ( T k + 1 ⋆ , T k ⋆ ] v\in(T^\star_{k+1},T^\star_k] v ∈ ( T k + 1 ⋆ , T k ⋆ ] X T k + 1 ⋆ ℓ k + 1 ⋆ = j k + 1 ⋆ X^{\ell^\star_{k+1}}_{T^\star_{k+1}}=j^\star_{k+1} X T k + 1 ⋆ ℓ k + 1 ⋆ = j k + 1 ⋆
Under the same non-explosion conditions as in the first-reaction construction, this defines the reverse CTMC on [ s , t ] [s,t] [ s , t ]
Gillespie discretization with partially frozen rates
Exact simulation with time-inhomogeneous Gillespie dynamics requires integrating the time-dependent transition rates and reevaluating their learned component after every one-position jump. To reduce the number of neural function evaluations, consider the grid 0 = t 0 < ⋯ < t n = 1 0=t_0<\cdots<t_n=1 0 = t 0 < ⋯ < t n = 1 s i : = t i − 1 s_i:=t_{i-1} s i := t i − 1 Q ˉ r ( ⋅ ; t i ∣ ⋅ ) \bar{Q}_r(\cdot;t_i\mid\cdot) Q ˉ r ( ⋅ ; t i ∣ ⋅ ) § 3.2 . On each interval [ s i , t i ] [s_i,t_i] [ s i , t i ] Q ˉ r θ ( ⋅ ; t i ∣ ⋅ ) \bar{Q}^{\theta}_r(\cdot;t_i\mid\cdot) Q ˉ r θ ( ⋅ ; t i ∣ ⋅ )
Equivalence of leave-one-out-frozen Gillespie and ancestral sampling
Fix [ s , t ] = [ s i , t i ] [s,t]=[s_i,t_i] [ s , t ] = [ s i , t i ] X t = x X_t=x X t = x p ^ i ℓ : = x ^ 0 θ ( x , t ) ℓ \hat{p}_i^\ell:=\hat{x}_0^\theta(x,t)^\ell p ^ i ℓ := x ^ 0 θ ( x , t ) ℓ (3) , using the bridge extension (17) is
p s ∣ t θ ( w ∣ x ) = ∏ ℓ = 1 L q s ∣ 0 , t ℓ ( w ℓ ∣ p ^ i ℓ , x ℓ ) = ∏ ℓ = 1 L q s ∣ 0 ℓ ( w ℓ ∣ p ^ i ℓ ) q t ∣ s ℓ ( x ℓ ∣ w ℓ ) q t ∣ 0 ℓ ( x ℓ ∣ p ^ i ℓ ) . (22) p_{s|t}^\theta(w|x)=\prod_{\ell=1}^{L}q_{s|0,t}^\ell(w^\ell|\hat{p}_i^\ell,x^\ell)
=\prod_{\ell=1}^{L}\frac{q_{s|0}^\ell(w^\ell|\hat{p}_i^\ell)q_{t|s}^\ell(x^\ell|w^\ell)}{q_{t|0}^\ell(x^\ell|\hat{p}_i^\ell)}. \tag*{(22)} p s ∣ t θ ( w ∣ x ) = ℓ = 1 ∏ L q s ∣0 , t ℓ ( w ℓ ∣ p ^ i ℓ , x ℓ ) = ℓ = 1 ∏ L q t ∣0 ℓ ( x ℓ ∣ p ^ i ℓ ) q s ∣0 ℓ ( w ℓ ∣ p ^ i ℓ ) q t ∣ s ℓ ( x ℓ ∣ w ℓ ) . (22) For r ∈ [ s , t ] r\in[s,t] r ∈ [ s , t ] y ≠ z ℓ y\ne z^\ell y = z ℓ
Q ‾ t θ , ℓ , l o o ( y ; t i ∣ z ) = Q t ℓ ( z ℓ ∣ y ) q r ∣ 0 ℓ ( y ∣ p ^ i ℓ ) q r ∣ 0 ℓ ( z ℓ ∣ p ^ i ℓ ) . (23) \overline{Q}_{t}^{\theta,\ell,\mathrm{loo}}(y;t_i|z)=Q_t^\ell(z^\ell|y)\frac{q_{r|0}^\ell(y|\hat{p}_i^\ell)}{q_{r|0}^\ell(z^\ell|\hat{p}_i^\ell)}. \tag*{(23)} Q t θ , ℓ , loo ( y ; t i ∣ z ) = Q t ℓ ( z ℓ ∣ y ) q r ∣0 ℓ ( z ℓ ∣ p ^ i ℓ ) q r ∣0 ℓ ( y ∣ p ^ i ℓ ) . (23) Since q r ∣ 0 ℓ ( y ∣ p ^ i ℓ ) q_{r|0}^\ell(y|\hat{p}_i^\ell) q r ∣0 ℓ ( y ∣ p ^ i ℓ ) r r r Cat ( p ^ i ℓ ) \operatorname{Cat}(\hat{p}_i^\ell) Cat ( p ^ i ℓ ) Q r ℓ ( z ℓ ∣ y ) Q_r^\ell(z^\ell|y) Q r ℓ ( z ℓ ∣ y ) (23) is exactly the time-reversed rates of this process. Therefore we know the transition from x ℓ x^\ell x ℓ t t t w ℓ w^\ell w ℓ s s s
q s ∣ 0 ℓ ( w ℓ ∣ p ^ i ℓ ) q t ∣ s ℓ ( x ℓ ∣ w ℓ ) q t ∣ 0 ℓ ( x ℓ ∣ p ^ i ℓ ) = q s ∣ 0 , t ℓ ( w ℓ ∣ p ^ i ℓ , x ℓ ) . \frac{q_{s|0}^\ell(w^\ell|\hat{p}_i^\ell)q_{t|s}^\ell(x^\ell|w^\ell)}{q_{t|0}^\ell(x^\ell|\hat{p}_i^\ell)}=q_{s|0,t}^\ell(w^\ell|\hat{p}_i^\ell,x^\ell). q t ∣0 ℓ ( x ℓ ∣ p ^ i ℓ ) q s ∣0 ℓ ( w ℓ ∣ p ^ i ℓ ) q t ∣ s ℓ ( x ℓ ∣ w ℓ ) = q s ∣0 , t ℓ ( w ℓ ∣ p ^ i ℓ , x ℓ ) . The frozen rates act independently across positions. Thus on the full joint distribution, exact Gillespie sampling gives
∏ ℓ = 1 L q s ∣ 0 , t ℓ ( w ℓ ∣ p ^ i ℓ , x ℓ ) = p s ∣ t θ ( w ∣ x ) . \prod_{\ell=1}^{L}q_{s|0,t}^\ell(w^\ell|\hat{p}_i^\ell,x^\ell)=p_{s|t}^\theta(w|x). ℓ = 1 ∏ L q s ∣0 , t ℓ ( w ℓ ∣ p ^ i ℓ , x ℓ ) = p s ∣ t θ ( w ∣ x ) . Therefore, the proposed discretization of Gillespie’s algorithm with leave-one-out frozen rates is equivalent to standard ancestral sampling used in the discrete-time setting. The same conclusion does not hold in general when freezing the score or the ordinary denoising posterior.
Euler and categorical tau-leaping
Token-wise Euler. Token-wise Euler freezes the state and time dependence of the rate matrix, effectively drawing the positions conditionally independently. For each ℓ \ell ℓ
P ( X s ℓ = y ∣ X t = x ) = { h Q ‾ t θ , ℓ ( y ∣ x ) y ≠ x ℓ , 1 + h Q ‾ t θ , ℓ ( x ℓ ∣ x ) y = x ℓ . (24) \mathbb{P}(X_s^\ell=y\mid X_t=x)=
\begin{cases}
h\overline{Q}_t^{\theta,\ell}(y|x) & y\ne x^\ell,\\
1+h\overline{Q}_t^{\theta,\ell}(x^\ell|x) & y=x^\ell.
\end{cases} \tag*{(24)} P ( X s ℓ = y ∣ X t = x ) = { h Q t θ , ℓ ( y ∣ x ) 1 + h Q t θ , ℓ ( x ℓ ∣ x ) y = x ℓ , y = x ℓ . (24) The step is valid whenever h ∑ y ≠ x ℓ Q ‾ t θ , ℓ ( y ∣ x ) ≤ 1 h\sum_{y\ne x^\ell}\overline{Q}_t^{\theta,\ell}(y|x)\le1 h ∑ y = x ℓ Q t θ , ℓ ( y ∣ x ) ≤ 1 ℓ \ell ℓ
Categorical τ \tau τ Categorical τ \tau τ
N i ℓ y ∼ Poisson ( h Q ‾ t θ , ℓ ( y ∣ x ) ) , ℓ ∈ [ 1 , L ] , y ≠ x ℓ . (25) N_i^{\ell y}\sim\operatorname{Poisson}\left(h\overline{Q}_t^{\theta,\ell}(y|x)\right),\qquad\ell\in[1,L],\quad y\ne x^\ell. \tag*{(25)} N i ℓ y ∼ Poisson ( h Q t θ , ℓ ( y ∣ x ) ) , ℓ ∈ [ 1 , L ] , y = x ℓ . (25) If ∑ y ≠ x ℓ N i ℓ y = 1 \sum_{y\ne x^\ell}N_i^{\ell y}=1 ∑ y = x ℓ N i ℓ y = 1 ℓ \ell ℓ y y y N i ℓ y = 1 N_i^{\ell y}=1 N i ℓ y = 1 x ℓ x^\ell x ℓ λ ˉ t θ , ℓ ( x ) : = ∑ y ≠ x ℓ Q ‾ t θ , ℓ ( y ∣ x ) \bar{\lambda}_t^{\theta,\ell}(x):=\sum_{y\ne x^\ell}\overline{Q}_t^{\theta,\ell}(y|x) λ ˉ t θ , ℓ ( x ) := ∑ y = x ℓ Q t θ , ℓ ( y ∣ x ) P ( X s ℓ = y ∣ X t = x ) = h Q ‾ t θ , ℓ ( y ∣ x ) exp { − h λ ˉ t θ , ℓ ( x ) } \mathbb{P}(X_s^\ell=y\mid X_t=x)=h\overline{Q}_t^{\theta,\ell}(y|x)\exp\{-h\bar{\lambda}_t^{\theta,\ell}(x)\} P ( X s ℓ = y ∣ X t = x ) = h Q t θ , ℓ ( y ∣ x ) exp { − h λ ˉ t θ , ℓ ( x )} y ≠ x ℓ y\ne x^\ell y = x ℓ x ℓ x^\ell x ℓ
Explicit partially frozen updates for MDM and UDM
Besides the score representation in (11) , the density ratio in (6) admits the identities
p t ℓ ( w ℓ ∣ x − ℓ ) p t ℓ ( x ℓ ∣ x − ℓ ) = ∑ x 0 ∈ V q t ∣ 0 ℓ ( w ℓ ∣ x 0 ) q t ∣ 0 ℓ ( x ℓ ∣ x 0 ) p 0 ∣ t ℓ ( x 0 ∣ x ) (26) \frac{p_t^\ell(w^\ell\mid x^{-\ell})}{p_t^\ell(x^\ell\mid x^{-\ell})}
=
\sum_{x_0\in\mathcal{V}}
\frac{q_{t\mid0}^\ell(w^\ell\mid x_0)}{q_{t\mid0}^\ell(x^\ell\mid x_0)}
p_{0\mid t}^\ell(x_0\mid x) \tag*{(26)} p t ℓ ( x ℓ ∣ x − ℓ ) p t ℓ ( w ℓ ∣ x − ℓ ) = x 0 ∈ V ∑ q t ∣ 0 ℓ ( x ℓ ∣ x 0 ) q t ∣ 0 ℓ ( w ℓ ∣ x 0 ) p 0 ∣ t ℓ ( x 0 ∣ x ) (26) p t ℓ ( w ℓ ∣ x − ℓ ) p t ℓ ( x ℓ ∣ x − ℓ ) = q t ∣ 0 ℓ ( w ℓ ∣ x ^ 0 ∣ t ℓ ( x ) ℓ ) q t ∣ 0 ℓ ( x ℓ ∣ x ^ 0 ∣ t ℓ ( x ) ℓ ) (27) \frac{p_t^\ell(w^\ell\mid x^{-\ell})}{p_t^\ell(x^\ell\mid x^{-\ell})}
=
\frac{q_{t\mid0}^\ell(w^\ell\mid\hat{x}_{0\mid t}^\ell(x)^\ell)}
{q_{t\mid0}^\ell(x^\ell\mid\hat{x}_{0\mid t}^\ell(x)^\ell)} \tag*{(27)} p t ℓ ( x ℓ ∣ x − ℓ ) p t ℓ ( w ℓ ∣ x − ℓ ) = q t ∣ 0 ℓ ( x ℓ ∣ x ^ 0 ∣ t ℓ ( x ) ℓ ) q t ∣ 0 ℓ ( w ℓ ∣ x ^ 0 ∣ t ℓ ( x ) ℓ ) (27) These identities suggest three freezing schemes, by freezing the score in (11) , the denoiser in (26) or the leave-one-out denoiser in (27) . We give their explicit forms below.
MDM. Since π ℓ = m \pi^\ell=m π ℓ = m (4) gives Q v ℓ ( z ℓ ∣ y ) = ( − α ˙ v / α v ) 1 z ℓ = m Q_v^\ell(z^\ell\mid y)=(-\dot{\alpha}_v/\alpha_v)\mathbf{1}_{z^\ell=m} Q v ℓ ( z ℓ ∣ y ) = ( − α ˙ v / α v ) 1 z ℓ = m y ≠ z ℓ y\ne z^\ell y = z ℓ (6) permits a reverse jump only from z ℓ = m z^\ell=m z ℓ = m y ≠ m y\ne m y = m
Q ˉ v θ , ℓ ( y ; t i ∣ z ) = ⟨ y ~ , s t i θ ( X t i ) ℓ ⟩ ( − α ˙ v α v ) , ∫ u r Q ˉ v θ , ℓ ( y ; t i ∣ z ) d v = ⟨ y ~ , s t i θ ( X t i ) ℓ ⟩ log ( α u α r ) . \bar{Q}_v^{\theta,\ell}(y;t_i\mid z)
=
\langle\tilde{y},s_{t_i}^{\theta}(X_{t_i})^\ell\rangle
\left(-\frac{\dot{\alpha}_v}{\alpha_v}\right),
\qquad
\int_u^r \bar{Q}_v^{\theta,\ell}(y;t_i\mid z)\,\mathrm{d}v
=
\langle\tilde{y},s_{t_i}^{\theta}(X_{t_i})^\ell\rangle
\log\left(\frac{\alpha_u}{\alpha_r}\right). Q ˉ v θ , ℓ ( y ; t i ∣ z ) = ⟨ y ~ , s t i θ ( X t i ) ℓ ⟩ ( − α v α ˙ v ) , ∫ u r Q ˉ v θ , ℓ ( y ; t i ∣ z ) d v = ⟨ y ~ , s t i θ ( X t i ) ℓ ⟩ log ( α r α u ) . Under denoising-posterior freezing, they are
Q ˉ v θ , ℓ ( y ; t i ∣ z ) = p 0 ∣ t i θ , ℓ ( y ∣ X t i ) ( − α ˙ v 1 − α v ) , ∫ u r Q ˉ v θ , ℓ ( y ; t i ∣ z ) d v = p 0 ∣ t i θ , ℓ ( y ∣ X t i ) log ( 1 − α r 1 − α u ) . \bar{Q}_v^{\theta,\ell}(y;t_i\mid z)
=
p_{0\mid t_i}^{\theta,\ell}(y\mid X_{t_i})
\left(-\frac{\dot{\alpha}_v}{1-\alpha_v}\right),
\qquad
\int_u^r \bar{Q}_v^{\theta,\ell}(y;t_i\mid z)\,\mathrm{d}v
=
p_{0\mid t_i}^{\theta,\ell}(y\mid X_{t_i})
\log\left(\frac{1-\alpha_r}{1-\alpha_u}\right). Q ˉ v θ , ℓ ( y ; t i ∣ z ) = p 0 ∣ t i θ , ℓ ( y ∣ X t i ) ( − 1 − α v α ˙ v ) , ∫ u r Q ˉ v θ , ℓ ( y ; t i ∣ z ) d v = p 0 ∣ t i θ , ℓ ( y ∣ X t i ) log ( 1 − α u 1 − α r ) . For masked diffusion, leave-one-out-denoiser and denoiser coincide on non-masked tokens so the rate and its integral have the same form as under denoising-posterior freezing.
UDM. Under score freezing, every y ≠ z ℓ y\ne z^\ell y = z ℓ
Q ˉ v θ , ℓ ( y ; t i ∣ z ) = ⟨ y ~ , s t i θ ( X t i ) ℓ ⟩ K ( − α ˙ v α v ) , ∫ u r Q ˉ v θ , ℓ ( y ; t i ∣ z ) d v = ⟨ y ~ , s t i θ ( X t i ) ℓ ⟩ K log ( α u α r ) . \bar{Q}_v^{\theta,\ell}(y;t_i\mid z)
=
\frac{\langle\tilde{y},s_{t_i}^{\theta}(X_{t_i})^\ell\rangle}{K}
\left(-\frac{\dot{\alpha}_v}{\alpha_v}\right),
\qquad
\int_u^r \bar{Q}_v^{\theta,\ell}(y;t_i\mid z)\,\mathrm{d}v
=
\frac{\langle\tilde{y},s_{t_i}^{\theta}(X_{t_i})^\ell\rangle}{K}
\log\left(\frac{\alpha_u}{\alpha_r}\right). Q ˉ v θ , ℓ ( y ; t i ∣ z ) = K ⟨ y ~ , s t i θ ( X t i ) ℓ ⟩ ( − α v α ˙ v ) , ∫ u r Q ˉ v θ , ℓ ( y ; t i ∣ z ) d v = K ⟨ y ~ , s t i θ ( X t i ) ℓ ⟩ log ( α r α u ) . For denoising-posterior freezing, the rate is
Q ˉ v θ , ℓ ( y ; t i ∣ z ) = 1 K ( − α ˙ v α v ) ( 1 − p 0 ∣ t i θ , ℓ ( y ∣ X t i ) − p 0 ∣ t i θ , ℓ ( z ℓ ∣ X t i ) + p 0 ∣ t i θ , ℓ ( y ∣ X t i ) 1 + ( K − 1 ) α v 1 − α v + p 0 ∣ t i θ , ℓ ( z ℓ ∣ X t i ) 1 − α v 1 + ( K − 1 ) α v ) , \bar{Q}_v^{\theta,\ell}(y;t_i\mid z)
=
\frac{1}{K}\left(-\frac{\dot{\alpha}_v}{\alpha_v}\right)
\left(
1-p_{0\mid t_i}^{\theta,\ell}(y\mid X_{t_i})-p_{0\mid t_i}^{\theta,\ell}(z^\ell\mid X_{t_i})
+p_{0\mid t_i}^{\theta,\ell}(y\mid X_{t_i})\frac{1+(K-1)\alpha_v}{1-\alpha_v}
+p_{0\mid t_i}^{\theta,\ell}(z^\ell\mid X_{t_i})\frac{1-\alpha_v}{1+(K-1)\alpha_v}
\right), Q ˉ v θ , ℓ ( y ; t i ∣ z ) = K 1 ( − α v α ˙ v ) ( 1 − p 0 ∣ t i θ , ℓ ( y ∣ X t i ) − p 0 ∣ t i θ , ℓ ( z ℓ ∣ X t i ) + p 0 ∣ t i θ , ℓ ( y ∣ X t i ) 1 − α v 1 + ( K − 1 ) α v + p 0 ∣ t i θ , ℓ ( z ℓ ∣ X t i ) 1 + ( K − 1 ) α v 1 − α v ) , whose integral is
∫ u r Q ˉ v θ , ℓ ( y ; t i ∣ z ) d v = 1 K log ( α u α r ) + p 0 ∣ t i θ , ℓ ( y ∣ X t i ) log ( 1 − α r 1 − α u ) + p 0 ∣ t i θ , ℓ ( z ℓ ∣ X t i ) K − 1 log ( 1 + ( K − 1 ) α r 1 + ( K − 1 ) α u ) . \int_u^r \bar{Q}_v^{\theta,\ell}(y;t_i\mid z)\,\mathrm{d}v
=
\frac{1}{K}\log\left(\frac{\alpha_u}{\alpha_r}\right)
+p_{0\mid t_i}^{\theta,\ell}(y\mid X_{t_i})\log\left(\frac{1-\alpha_r}{1-\alpha_u}\right)
+\frac{p_{0\mid t_i}^{\theta,\ell}(z^\ell\mid X_{t_i})}{K-1}
\log\left(\frac{1+(K-1)\alpha_r}{1+(K-1)\alpha_u}\right). ∫ u r Q ˉ v θ , ℓ ( y ; t i ∣ z ) d v = K 1 log ( α r α u ) + p 0 ∣ t i θ , ℓ ( y ∣ X t i ) log ( 1 − α u 1 − α r ) + K − 1 p 0 ∣ t i θ , ℓ ( z ℓ ∣ X t i ) log ( 1 + ( K − 1 ) α u 1 + ( K − 1 ) α r ) . Finally, under leave-one-out-denoiser freezing,
Q ˉ v θ , ℓ ( y ; t i ∣ z ) = 1 K ( − α ˙ v α v ) α v ⟨ y ~ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ + ( 1 − α v ) / K α v ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ + ( 1 − α v ) / K , \bar{Q}_v^{\theta,\ell}(y;t_i\mid z)
=
\frac{1}{K}\left(-\frac{\dot{\alpha}_v}{\alpha_v}\right)
\frac{\alpha_v\langle\tilde{y},\hat{x}_0^\theta(X_{t_i},t_i)^\ell\rangle+(1-\alpha_v)/K}
{\alpha_v\langle\tilde{z}^\ell,\hat{x}_0^\theta(X_{t_i},t_i)^\ell\rangle+(1-\alpha_v)/K}, Q ˉ v θ , ℓ ( y ; t i ∣ z ) = K 1 ( − α v α ˙ v ) α v ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ + ( 1 − α v ) / K α v ⟨ y ~ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ + ( 1 − α v ) / K , ∫ u r Q ˉ v θ , ℓ ( y ; t i ∣ z ) d v = 1 K log ( α u α r ) + ⟨ y ~ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ K ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − 1 log ( 1 + ( K ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − 1 ) α u 1 + ( K ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − 1 ) α r ) . \begin{aligned}
\int_u^r \bar{Q}_v^{\theta,\ell}(y;t_i\mid z)\,\mathrm{d}v
={}&
\frac{1}{K}\log\left(\frac{\alpha_u}{\alpha_r}\right)\\
&+
\frac{\langle\tilde{y},\hat{x}_0^\theta(X_{t_i},t_i)^\ell\rangle-\langle\tilde{z}^\ell,\hat{x}_0^\theta(X_{t_i},t_i)^\ell\rangle}
{K\langle\tilde{z}^\ell,\hat{x}_0^\theta(X_{t_i},t_i)^\ell\rangle-1}
\log\left(
\frac{1+(K\langle\tilde{z}^\ell,\hat{x}_0^\theta(X_{t_i},t_i)^\ell\rangle-1)\alpha_u}
{1+(K\langle\tilde{z}^\ell,\hat{x}_0^\theta(X_{t_i},t_i)^\ell\rangle-1)\alpha_r}
\right).
\end{aligned} ∫ u r Q ˉ v θ , ℓ ( y ; t i ∣ z ) d v = K 1 log ( α r α u ) + K ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − 1 ⟨ y ~ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ log ( 1 + ( K ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − 1 ) α r 1 + ( K ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − 1 ) α u ) . When K ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − 1 = 0 K\langle\tilde{z}^\ell,\hat{x}_0^\theta(X_{t_i},t_i)^\ell\rangle-1=0 K ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − 1 = 0
( ⟨ y ~ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ ) ( α u − α r ) . \left(\langle\tilde{y},\hat{x}_0^\theta(X_{t_i},t_i)^\ell\rangle-\langle\tilde{z}^\ell,\hat{x}_0^\theta(X_{t_i},t_i)^\ell\rangle\right)(\alpha_u-\alpha_r). ( ⟨ y ~ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ − ⟨ z ~ ℓ , x ^ 0 θ ( X t i , t i ) ℓ ⟩ ) ( α u − α r ) . After every UDM transition, z ℓ z^{\ell} z ℓ
From integrated rates to jump updates
For each candidate transition ( ℓ , y ) (\ell,y) ( ℓ , y ) T ℓ y T^{\ell y} T ℓ y [ s , r ] [s,r] [ s , r ]
∫ T ℓ y r Q ~ v θ , ℓ ( y ; t i ∣ z ) d v = E r ℓ y . (28) \int_{T^{\ell y}}^{r} \widetilde{Q}^{\theta,\ell}_{v}(y;t_i\mid z)\,\mathrm{d}v = E^{\ell y}_{r}. \tag*{(28)} ∫ T ℓ y r Q v θ , ℓ ( y ; t i ∣ z ) d v = E r ℓ y . (28) If the integrated rate over [ s , r ] [s,r] [ s , r ] E r ℓ y E^{\ell y}_{r} E r ℓ y T ℓ y = − ∞ T^{\ell y}=-\infty T ℓ y = − ∞
Closed-form update for MDM. All active rates share the same time-dependent factor, so we select the smallest normalized residual and evaluate α − 1 \alpha^{-1} α − 1
Throughout this paragraph, a quotient with zero denominator is understood as + ∞ +\infty + ∞
Under score freezing, select
( ℓ ^ , y ^ ) ∈ a r g m i n ℓ : ℓ ≠ m y ≠ m E r ℓ y ⟨ y ~ , s t i θ , ℓ ( X t i ) ℓ ⟩ . (\hat{\ell},\hat{y})\in\mathop{\mathrm{argmin}}_{\substack{\ell:\ell\ne m\\y\ne m}}\frac{E^{\ell y}_{r}}{\langle\tilde{y},s^{\theta,\ell}_{t_i}(X_{t_i})^{\ell}\rangle}. ( ℓ ^ , y ^ ) ∈ argmin ℓ : ℓ = m y = m ⟨ y ~ , s t i θ , ℓ ( X t i ) ℓ ⟩ E r ℓ y . If the minimum exceeds log ( α s / α r ) \log(\alpha_s/\alpha_r) log ( α s / α r ) s s s
T ⋆ = α − 1 ( α r exp { E r ℓ ^ y ^ ⟨ y ^ , s t i θ , ℓ ^ ( X t i ) ℓ ^ ⟩ } ) . T^\star=\alpha^{-1}\left(\alpha_r\exp\left\{\frac{E^{\hat{\ell}\hat{y}}_{r}}{\langle\hat{y},s^{\theta,\hat{\ell}}_{t_i}(X_{t_i})^{\hat{\ell}}\rangle}\right\}\right). T ⋆ = α − 1 ( α r exp { ⟨ y ^ , s t i θ , ℓ ^ ( X t i ) ℓ ^ ⟩ E r ℓ ^ y ^ } ) . For denoising-posterior freezing, the selection and inversion have the form in Appendix § 10 , with its cached denoiser probability replaced by p 0 ∣ t i θ , ℓ ( y ∣ X t i ) p^{\theta,\ell}_{0\mid t_i}(y\mid X_{t_i}) p 0 ∣ t i θ , ℓ ( y ∣ X t i )
Root finding for UDM. Score-frozen UDM candidate times are also explicit. The candidate time for ( ℓ , y ) (\ell,y) ( ℓ , y ) [ s , r ] [s,r] [ s , r ]
⟨ y ~ , s t i θ , ℓ ( X t i ) ℓ ⟩ K log ( α s α r ) ≥ E r ℓ y , \frac{\langle\tilde{y},s^{\theta,\ell}_{t_i}(X_{t_i})^{\ell}\rangle}{K}\log\left(\frac{\alpha_s}{\alpha_r}\right)\ge E^{\ell y}_{r}, K ⟨ y ~ , s t i θ , ℓ ( X t i ) ℓ ⟩ log ( α r α s ) ≥ E r ℓ y , and, when its coefficient is positive,
T ℓ y = α − 1 ( α r exp { K E r ℓ y ⟨ y ~ , s t i θ , ℓ ( X t i ) ℓ ⟩ } ) . T^{\ell y}=\alpha^{-1}\left(\alpha_r\exp\left\{\frac{K E^{\ell y}_{r}}{\langle\tilde{y},s^{\theta,\ell}_{t_i}(X_{t_i})^{\ell}\rangle}\right\}\right). T ℓ y = α − 1 ( α r exp { ⟨ y ~ , s t i θ , ℓ ( X t i ) ℓ ⟩ K E r ℓ y } ) . For denoising-posterior and leave-one-out-denoiser freezing, (28) has no closed-form solution in general so we solve the equation with bisection. If the integrated rate over [ s , r ] [s,r] [ s , r ] E r ℓ y E^{\ell y}_{r} E r ℓ y [ s , r ] [s,r] [ s , r ] m m m m m m ∫ m r Q ~ v θ , ℓ ( y ; t i ∣ z ) d v ≥ E r ℓ y \int_{m}^{r}\widetilde{Q}^{\theta,\ell}_{v}(y;t_i\mid z)\,\mathrm{d}v\ge E^{\ell y}_{r} ∫ m r Q v θ , ℓ ( y ; t i ∣ z ) d v ≥ E r ℓ y m m m m m m
Exactness of leave-one-out freezing under factorization
Suppose that the data distribution factorizes across positions, and write p 0 ℓ : = E [ X ~ 0 ℓ ] ∈ Δ K p^\ell_0:=\mathbb{E}[\widetilde{X}^{\ell}_0]\in\Delta_K p 0 ℓ := E [ X 0 ℓ ] ∈ Δ K ℓ \ell ℓ r r r ℓ \ell ℓ q r ℓ ( ⋅ ∣ p 0 ℓ ) q^\ell_r(\cdot\mid p^\ell_0) q r ℓ ( ⋅ ∣ p 0 ℓ )
x ^ 0 ∣ r L O O ( z ) ℓ = E [ X ~ 0 ℓ ∣ X r − ℓ = z − ℓ ] = p 0 ℓ \hat{x}^{\mathrm{LOO}}_{0\mid r}(z)^\ell=\mathbb{E}[\widetilde{X}^{\ell}_0\mid X^{- \ell}_r=z^{-\ell}]=p^\ell_0 x ^ 0 ∣ r LOO ( z ) ℓ = E [ X 0 ℓ ∣ X r − ℓ = z − ℓ ] = p 0 ℓ for every r r r z z z ( t , x ) (t,x) ( t , x ) z z z w w w ℓ \ell ℓ
Q ‾ r θ , l o o ( w ; t i ∣ z ) = Q r ℓ ( z ℓ ∣ w ℓ ) q t i ℓ ( w ℓ ∣ p 0 ℓ ) q t i ℓ ( z ℓ ∣ p 0 ℓ ) = Q ‾ r ( w ∣ z ) . (29) \overline{Q}^{\theta,\mathrm{loo}}_r(w;t_i\mid z)=Q^\ell_r(z^\ell\mid w^\ell)\frac{q^\ell_{t_i}(w^\ell\mid p^\ell_0)}{q^\ell_{t_i}(z^\ell\mid p^\ell_0)}=\overline{Q}_r(w\mid z). \tag*{(29)} Q r θ , loo ( w ; t i ∣ z ) = Q r ℓ ( z ℓ ∣ w ℓ ) q t i ℓ ( z ℓ ∣ p 0 ℓ ) q t i ℓ ( w ℓ ∣ p 0 ℓ ) = Q r ( w ∣ z ) . (29) Hence the partially frozen generator is the exact reverse generator throughout every discretization interval, and its exact integration introduces no discretization error. The single-token case is the special case L = 1 L=1 L = 1 p 0 p_0 p 0
The other potential frozen rate formulas, for score freezing and denoiser freezing, are given in Appendix § 12 .§ 5 : they retain either s t i θ ( x ) ℓ s^\theta_{t_i}(x)^\ell s t i θ ( x ) ℓ p 0 ∣ t i θ ( ⋅ ∣ x ) p^\theta_{0\mid t_i}(\cdot\mid x) p 0 ∣ t i θ ( ⋅ ∣ x ) r r r p 0 p_0 p 0 p 0 ( 1 ) = ρ ∈ ( 0 , 1 ) ∖ { 1 / 2 } p_0(1)=\rho\in(0,1)\setminus\{1/2\} p 0 ( 1 ) = ρ ∈ ( 0 , 1 ) ∖ { 1/2 } p r ( 1 ) = α r ρ + ( 1 − α r ) / 2 p_r(1)=\alpha_r\rho+(1-\alpha_r)/2 p r ( 1 ) = α r ρ + ( 1 − α r ) /2
⟨ 2 ~ , s r ( 1 ) 1 ⟩ = p r ( 2 ) p r ( 1 ) , p 0 ∣ r 1 ( 1 ∣ 1 ) = ρ ( 1 + α r ) 2 p r ( 1 ) . \langle\widetilde{2},s_r(1)^1\rangle=\frac{p_r(2)}{p_r(1)},\qquad p^1_{0\mid r}(1\mid1)=\frac{\rho(1+\alpha_r)}{2p_r(1)}. ⟨ 2 , s r ( 1 ) 1 ⟩ = p r ( 1 ) p r ( 2 ) , p 0 ∣ r 1 ( 1 ∣ 1 ) = 2 p r ( 1 ) ρ ( 1 + α r ) . Both vary with α r \alpha_r α r t t t r < t r<t r < t p 0 ℓ p^\ell_0 p 0 ℓ
A counterexample for score freezing
We now show that freezing the score for the ERF procedure while keeping the original forward rates can lead to an infinite number of jumps in finite time. Consider UDM with one position, two tokens, and the schedule α t = exp ( − 2 t ) \alpha_t=\exp(-2t) α t = exp ( − 2 t )
Q t ( 2 ∣ 1 ) = Q t ( 1 ∣ 2 ) = 1. Q_t(2\mid1)=Q_t(1\mid2)=1. Q t ( 2 ∣ 1 ) = Q t ( 1 ∣ 2 ) = 1. Freeze the score at time t i = 1 / 10 t_i=1/10 t i = 1/10 X t i = 1 X_{t_i}=1 X t i = 1 s t i ( 1 ) = ( 1 , 2 ) s_{t_i}(1)=(1,2) s t i ( 1 ) = ( 1 , 2 )
Q ‾ t θ ( 2 ; t i ∣ 1 ) = 2 , Q ‾ t θ ( 1 ; t i ∣ 2 ) = 1. \overline{Q}^{\theta}_t(2;t_i\mid1)=2,\qquad\overline{Q}^{\theta}_t(1;t_i\mid2)=1. Q t θ ( 2 ; t i ∣ 1 ) = 2 , Q t θ ( 1 ; t i ∣ 2 ) = 1. The ERF nevertheless retains the original forward exit rate, equal to one, as the passive speed in either token. These active and passive speeds are incompatible: if the frozen reverse rates were the time reversal of the original forward rates under a positive law q t q_t q t q t ( 2 ) / q t ( 1 ) = 2 q_t(2)/q_t(1)=2 q t ( 2 ) / q t ( 1 ) = 2 q t ( 1 ) / q t ( 2 ) = 1 q_t(1)/q_t(2)=1 q t ( 1 ) / q t ( 2 ) = 1
Let ( E t i 11 , E t i 12 ) = ( A , B ) (E^{11}_{t_i},E^{12}_{t_i})=(A,B) ( E t i 11 , E t i 12 ) = ( A , B ) A A A B B B Exp ( 1 ) \operatorname{Exp}(1) Exp ( 1 ) X t i = 1 X_{t_i}=1 X t i = 1 S : = A + B / 2 S:=A+B/2 S := A + B /2 B / 2 B/2 B /2 ( S , 0 ) (S,0) ( S , 0 ) S S S ( 0 , S ) (0,S) ( 0 , S ) ( 0 , R ) (0,R) ( 0 , R ) R R R ( 0 , R / 2 ) (0,R/2) ( 0 , R /2 ) S , S / 2 , S / 4 , … S,S/2,S/4,\ldots S , S /2 , S /4 , …
T a c c = B 2 + S + ∑ n = 0 ∞ S 2 n = 3 A + 2 B < ∞ . (30) T_{\mathrm{acc}}=\frac{B}{2}+S+\sum_{n=0}^{\infty}\frac{S}{2^n}=3A+2B<\infty. \tag*{(30)} T acc = 2 B + S + n = 0 ∑ ∞ 2 n S = 3 A + 2 B < ∞. (30) For every h > 0 h>0 h > 0 3 A + 2 B < h 3A+2B<h 3 A + 2 B < h 1 − 3 exp ( − h / 3 ) + 2 exp ( − h / 2 ) > 0 1-3\exp(-h/3)+2\exp(-h/2)>0 1 − 3 exp ( − h /3 ) + 2 exp ( − h /2 ) > 0
The issue is not with the score approximation, the cached score can be population-optimal. Indeed, choose
p 0 ( 1 ) = 1 2 − exp ( 1 / 5 ) 6 , p 0 ( 2 ) = 1 2 + exp ( 1 / 5 ) 6 . p_0(1)=\frac{1}{2}-\frac{\exp(1/5)}{6},\qquad p_0(2)=\frac{1}{2}+\frac{\exp(1/5)}{6}. p 0 ( 1 ) = 2 1 − 6 exp ( 1/5 ) , p 0 ( 2 ) = 2 1 + 6 exp ( 1/5 ) . Then p t i ( 1 ) = 1 / 3 p_{t_i}(1)=1/3 p t i ( 1 ) = 1/3 p t i ( 2 ) = 2 / 3 p_{t_i}(2)=2/3 p t i ( 2 ) = 2/3 ( 1 , 2 ) (1,2) ( 1 , 2 ) X t i = 1 X_{t_i}=1 X t i = 1 3 A + 2 B < t i 3A+2B<t_i 3 A + 2 B < t i
Why the explosion cannot occur with leave-one-out freezing
We now show that the same phenomenon cannot happen with leave-one-out freezing. The key idea is to show that the leave-one-out-frozen reverse generator in (13) is actually the exact time reversal of the original forward generator started from a different law at time 0. Therefore, Theorem I.1 applies and we cannot have an infinite number of jumps. Let p ^ i ℓ : = x ^ 0 ℓ ( X t i , t i ) \widehat{p}^{\ell}_{i} := \widehat{x}^{\ell}_{0}(X_{t_i},t_i) p i ℓ := x 0 ℓ ( X t i , t i ) ℓ ∈ [ 1 , L ] \ell\in[1,L] ℓ ∈ [ 1 , L ]
p ^ r ∣ i ( z ) : = ∏ ℓ = 1 L q r ∣ 0 ℓ ( z ℓ ∣ p ^ i ℓ ) . \widehat{p}_{r|i}(z) := \prod_{\ell=1}^{L} q^{\ell}_{r|0}(z^{\ell}|\widehat{p}^{\ell}_{i}). p r ∣ i ( z ) := ℓ = 1 ∏ L q r ∣0 ℓ ( z ℓ ∣ p i ℓ ) . Each factor is a marginal of the original forward process started at p 0 = ∏ ℓ = 1 L p ^ i ℓ p_0 = \prod_{\ell=1}^{L} \widehat{p}^{\ell}_{i} p 0 = ∏ ℓ = 1 L p i ℓ p ^ r ∣ i \widehat{p}_{r|i} p r ∣ i p ^ i \widehat{p}_i p i z z z w w w ℓ \ell ℓ (13) becomes
Q ‾ r ℓ , l o o ( w ; t i ∣ z ) = p ^ r ∣ i ( w ) p ^ r ∣ i ( z ) Q r ( z ∣ w ) . \overline{Q}^{\ell,\mathrm{loo}}_{r}(w;t_i|z) = \frac{\widehat{p}_{r|i}(w)}{\widehat{p}_{r|i}(z)} Q_r(z|w). Q r ℓ , loo ( w ; t i ∣ z ) = p r ∣ i ( z ) p r ∣ i ( w ) Q r ( z ∣ w ) . Thus the frozen reverse rates are the time reversal of the original forward rates under p ^ r ∣ i \widehat{p}_{r|i} p r ∣ i λ r ℓ ( z ℓ ) \lambda^{\ell}_{r}(z^{\ell}) λ r ℓ ( z ℓ )
Fix the cached predictions and an interval [ s , t ] [s,t] [ s , t ] p ^ r ∣ i ( z ) > 0 \widehat{p}_{r|i}(z)>0 p r ∣ i ( z ) > 0 r ∈ [ s , t ] r\in[s,t] r ∈ [ s , t ] z ∈ X z\in\mathcal{X} z ∈ X B i ( z ) B_i(z) B i ( z ) e ∈ ( 0 , ∞ ) L K e\in(0,\infty)^{LK} e ∈ ( 0 , ∞ ) L K ( X t , E t ) = ( z , e ) (X_t,E_t)=(z,e) ( X t , E t ) = ( z , e ) [ s , t ] [s,t] [ s , t ] Theorem 3.1 ,
0 = ∑ z ∈ X p ^ t ∣ i ( z ) Exp ( 1 ) ⊗ L K ( B i ( z ) ) . 0 = \sum_{z\in\mathcal{X}} \widehat{p}_{t|i}(z)\operatorname{Exp}(1)^{\otimes LK}(B_i(z)). 0 = z ∈ X ∑ p t ∣ i ( z ) Exp ( 1 ) ⊗ L K ( B i ( z )) . Each B i ( z ) B_i(z) B i ( z ) p ^ t ∣ i ( z ) > 0 \widehat{p}_{t|i}(z)>0 p t ∣ i ( z ) > 0 ( 0 , ∞ ) L K (0,\infty)^{LK} ( 0 , ∞ ) L K X t X_t X t p ^ t ∣ i \widehat{p}_{t|i} p t ∣ i E t E_t E t X t = z X_t=z X t = z z z z B i ( z ) B_i(z) B i ( z )
Net-flux rate matrix
Net flux construction and relation to prior work
The relevant decomposition is that of the probability flux induced by the reverse rates, rather than of the rates alone. Assume that p t ( x ) > 0 p_t(x)>0 p t ( x ) > 0 x ≠ w x\ne w x = w
F t ( x , w ) : = p t ( x ) Q ‾ t ( w ∣ x ) , S t ( x , w ) : = F t ( x , w ) + F t ( w , x ) 2 , J t ( x , w ) : = F t ( x , w ) − F t ( w , x ) . F_t(x,w) := p_t(x)\overline{Q}_t(w|x),\qquad S_t(x,w) := \frac{F_t(x,w)+F_t(w,x)}{2},\qquad J_t(x,w) := F_t(x,w)-F_t(w,x). F t ( x , w ) := p t ( x ) Q t ( w ∣ x ) , S t ( x , w ) := 2 F t ( x , w ) + F t ( w , x ) , J t ( x , w ) := F t ( x , w ) − F t ( w , x ) . Here S t S_t S t { x , w } \{x,w\} { x , w } J t J_t J t Q ‾ t ( x ∣ x ) = − ∑ w ≠ x Q ‾ t ( w ∣ x ) \overline{Q}_t(x|x)=-\sum_{w\ne x}\overline{Q}_t(w|x) Q t ( x ∣ x ) = − ∑ w = x Q t ( w ∣ x )
∂ t p t ( x ) = − ∑ w ≠ x p t ( w ) Q ‾ t ( x ∣ w ) = ∑ w ≠ x ( F t ( x , w ) − F t ( w , x ) ) = ∑ w ≠ x J t ( x , w ) . \partial_t p_t(x) = -\sum_{w\ne x}p_t(w)\overline{Q}_t(x|w) = \sum_{w\ne x}\bigl(F_t(x,w)-F_t(w,x)\bigr) = \sum_{w\ne x}J_t(x,w). ∂ t p t ( x ) = − w = x ∑ p t ( w ) Q t ( x ∣ w ) = w = x ∑ ( F t ( x , w ) − F t ( w , x ) ) = w = x ∑ J t ( x , w ) . Thus the marginal evolution depends only on J t J_t J t S t S_t S t F t ( x , w ) − F t ( w , x ) F_t(x,w)-F_t(w,x) F t ( x , w ) − F t ( w , x ) F t N F ( x , w ) = [ J t ( x , w ) ] + F_t^{\mathrm{NF}}(x,w)=[J_t(x,w)]_+ F t NF ( x , w ) = [ J t ( x , w ) ] +
Q ‾ t N F ( w ∣ x ) : = [ J t ( x , w ) ] + p t ( x ) = [ p t ( x ) Q ‾ t ( w ∣ x ) − p t ( w ) Q ‾ t ( x ∣ w ) ] + p t ( x ) , \overline{Q}^{\mathrm{NF}}_t(w|x) := \frac{[J_t(x,w)]_+}{p_t(x)} = \frac{\bigl[p_t(x)\overline{Q}_t(w|x)-p_t(w)\overline{Q}_t(x|w)\bigr]_+}{p_t(x)}, Q t NF ( w ∣ x ) := p t ( x ) [ J t ( x , w ) ] + = p t ( x ) [ p t ( x ) Q t ( w ∣ x ) − p t ( w ) Q t ( x ∣ w ) ] + , with diagonal entries chosen so that each row sums to zero. Since a + − ( − a ) + = a a_+ - (-a)_+ = a a + − ( − a ) + = a J t J_t J t Q ‾ t N F ( w ∣ x ) Q ‾ t N F ( x ∣ w ) = 0 \overline{Q}^{\mathrm{NF}}_t(w|x)\overline{Q}^{\mathrm{NF}}_t(x|w)=0 Q t NF ( w ∣ x ) Q t NF ( x ∣ w ) = 0 J t ( x , w ) J_t(x,w) J t ( x , w ) ∣ J t ( x , w ) ∣ |J_t(x,w)| ∣ J t ( x , w ) ∣
The time-reversal identity gives p t ( w ) Q ‾ t ( x ∣ w ) = p t ( x ) Q t ( w ∣ x ) p_t(w)\overline{Q}_t(x|w)=p_t(x)Q_t(w|x) p t ( w ) Q t ( x ∣ w ) = p t ( x ) Q t ( w ∣ x ) Q ‾ t N F ( w ∣ x ) = [ Q t ( w ∣ x ) − Q ‾ t ( w ∣ x ) ] + \overline{Q}^{\mathrm{NF}}_t(w|x)=[Q_t(w|x)-\overline{Q}_t(w|x)]_+ Q t NF ( w ∣ x ) = [ Q t ( w ∣ x ) − Q t ( w ∣ x ) ] +
A similar construction appears in Campbell et al. [4 ] , however the two constructions are not equivalent and ours leads to an even less stochastic process. Let us write Q ‾ t X 0 ( ⋅ ∣ ⋅ ) \overline{Q}^{X_0}_t(\cdot|\cdot) Q t X 0 ( ⋅ ∣ ⋅ ) [4 ] applies net flux conditionally and then averages under the posterior of X 0 X_0 X 0 x ≠ w x\ne w x = w
E [ Q ‾ t N F , X 0 ( w ∣ x ) ∣ X t = x ] and Q ‾ t N F ( w ∣ x ) , \mathbb{E}\left[\overline{Q}^{\mathrm{NF},X_0}_t(w|x)\mid X_t=x\right]\qquad\text{and}\qquad\overline{Q}^{\mathrm{NF}}_t(w|x), E [ Q t NF , X 0 ( w ∣ x ) ∣ X t = x ] and Q t NF ( w ∣ x ) , and they are not equal in general because the positive-part operation in the net-flux projection does not commute with averaging. More precisely, its convexity gives
Q ‾ t N F ( w ∣ x ) ≤ E [ Q ‾ t N F , X 0 ( w ∣ x ) ∣ X t = x ] . \overline{Q}^{\mathrm{NF}}_t(w|x)\le\mathbb{E}\left[\overline{Q}^{\mathrm{NF},X_0}_t(w|x)\mid X_t=x\right]. Q t NF ( w ∣ x ) ≤ E [ Q t NF , X 0 ( w ∣ x ) ∣ X t = x ] . The inequality is strict when different clean endpoints induce opposing currents between x x x w w w
Forward net-flux rates and passive residuals
ERF requires the forward rates associated with the chosen reverse generator to evolve its passive residuals. For strictly positive marginals and exact canonical rates, time reversal gives, for x ≠ w x\ne w x = w
Q t N F ( w ∣ x ) : = p t ( w ) p t ( x ) Q ‾ t N F ( x ∣ w ) = [ Q t ( w ∣ x ) − Q ‾ t ( w ∣ x ) ] + . (31) Q^{\mathrm{NF}}_t(w|x):=\frac{p_t(w)}{p_t(x)}\overline{Q}^{\mathrm{NF}}_t(x|w)=\left[Q_t(w|x)-\overline{Q}_t(w|x)\right]_+. \tag*{(31)} Q t NF ( w ∣ x ) := p t ( x ) p t ( w ) Q t NF ( x ∣ w ) = [ Q t ( w ∣ x ) − Q t ( w ∣ x ) ] + . (31) Indeed, substituting (15) gives [ p t ( w ) Q ‾ t ( x ∣ w ) − p t ( x ) Q ‾ t ( w ∣ x ) ] + / p t ( x ) [p_t(w)\overline{Q}_t(x|w)-p_t(x)\overline{Q}_t(w|x)]_+/p_t(x) [ p t ( w ) Q t ( x ∣ w ) − p t ( x ) Q t ( w ∣ x ) ] + / p t ( x ) Q t ( w ∣ x ) Q_t(w|x) Q t ( w ∣ x )
For position ℓ \ell ℓ b t N F , ℓ ( x ) : = ∑ w : w − ℓ = x − ℓ , w ℓ ≠ x ℓ Q t N F ( w ∣ x ) b^{\mathrm{NF},\ell}_t(x):=\sum_{w:w^{-\ell}=x^{-\ell},\,w^\ell\ne x^\ell}Q^{\mathrm{NF}}_t(w|x) b t NF , ℓ ( x ) := ∑ w : w − ℓ = x − ℓ , w ℓ = x ℓ Q t NF ( w ∣ x ) § 10 .
Score and denoising-posterior freezing for UDM net flux
Fix [ s , t ] = [ s i , t i ] [s,t]=[s_i,t_i] [ s , t ] = [ s i , t i ] x = X t i x=X_{t_i} x = X t i z z z r ∈ [ s , t ] r\in[s,t] r ∈ [ s , t ] y ≠ z ℓ y\ne z^\ell y = z ℓ − α ˙ v / ( K α v ) -\dot{\alpha}_v/(K\alpha_v) − α ˙ v / ( K α v ) § 12 .§ 5 gives the following off-diagonal rates.
Under score freezing, the projected reverse rate and the complementary passive forward exit rate at position ℓ \ell ℓ
Q ‾ v N F , θ , ℓ ( y ; t i ∣ z ) = ( − α ˙ v K α v ) [ y ~ , s t i θ ( x ) ℓ − 1 ] + , \overline{Q}^{\mathrm{NF},\theta,\ell}_v(y;t_i|z)=\left(-\frac{\dot{\alpha}_v}{K\alpha_v}\right)\left[\widetilde{y},s^\theta_{t_i}(x)^\ell-1\right]_+, Q v NF , θ , ℓ ( y ; t i ∣ z ) = ( − K α v α ˙ v ) [ y , s t i θ ( x ) ℓ − 1 ] + , b v N F , ℓ ( z ) = ( − α ˙ v K α v ) ∑ y ∈ V ∖ { z ℓ } [ 1 − ⟨ y ~ , s t i θ ( x ) ℓ ⟩ ] + . b^{\mathrm{NF},\ell}_v(z)=\left(-\frac{\dot{\alpha}_v}{K\alpha_v}\right)\sum_{y\in\mathcal{V}\setminus\{z^\ell\}}\left[1-\langle\widetilde{y},s^\theta_{t_i}(x)^\ell\rangle\right]_+. b v NF , ℓ ( z ) = ( − K α v α ˙ v ) y ∈ V ∖ { z ℓ } ∑ [ 1 − ⟨ y , s t i θ ( x ) ℓ ⟩ ] + . Under denoising-posterior freezing, the corresponding rates are
Q ‾ v N F , θ , ℓ ( y ; t i ∣ z ) = − α ˙ v [ p 0 ∣ t i θ , ℓ ( y ∣ x ) 1 − α v − p 0 ∣ t i θ , ℓ ( z ℓ ∣ x ) 1 + ( K − 1 ) α v ] + \overline{Q}^{\mathrm{NF},\theta,\ell}_v(y;t_i|z)=-\dot{\alpha}_v\left[\frac{p^{\theta,\ell}_{0|t_i}(y|x)}{1-\alpha_v}-\frac{p^{\theta,\ell}_{0|t_i}(z^\ell|x)}{1+(K-1)\alpha_v}\right]_+ Q v NF , θ , ℓ ( y ; t i ∣ z ) = − α ˙ v [ 1 − α v p 0∣ t i θ , ℓ ( y ∣ x ) − 1 + ( K − 1 ) α v p 0∣ t i θ , ℓ ( z ℓ ∣ x ) ] + b v N F , ℓ ( z ) = − α ˙ v ∑ y ∈ V ∖ { z ℓ } [ p 0 ∣ t i θ , ℓ ( z ℓ ∣ x ) 1 + ( K − 1 ) α v − p 0 ∣ t i θ , ℓ ( y ∣ x ) 1 − α v ] + . b^{\mathrm{NF},\ell}_v(z)=-\dot{\alpha}_v\sum_{y\in\mathcal{V}\setminus\{z^\ell\}}\left[\frac{p^{\theta,\ell}_{0|t_i}(z^\ell|x)}{1+(K-1)\alpha_v}-\frac{p^{\theta,\ell}_{0|t_i}(y|x)}{1-\alpha_v}\right]_+. b v NF , ℓ ( z ) = − α ˙ v y ∈ V ∖ { z ℓ } ∑ [ 1 + ( K − 1 ) α v p 0∣ t i θ , ℓ ( z ℓ ∣ x ) − 1 − α v p 0∣ t i θ , ℓ ( y ∣ x ) ] + . Each passive speed is the sum of the positive-part complements [ Q v ℓ ( y ∣ z ℓ ) − Q ‾ v θ , ℓ ( y ; t i ∣ z ) ] + [Q^\ell_v(y|z^\ell)-\overline{Q}^{\theta,\ell}_v(y;t_i|z)]_+ [ Q v ℓ ( y ∣ z ℓ ) − Q v θ , ℓ ( y ; t i ∣ z ) ] + y ≠ z ℓ y\ne z^\ell y = z ℓ (31) . The diagonal entries are the negatives of the corresponding row sums. The score-frozen rates integrate directly, while each expression inside the positive part of a posterior-frozen rate changes sign at most once as α v \alpha_v α v
Forward rates compatibility. The new forward rates justify the ERF construction only if they evolve a positive distribution under which the frozen reverse rates are their time reversals. This need not hold for score or denoising-posterior freezing: in the binary score example of Appendix § 12 .5.3, the net-flux reverse rate from token 1 to token 2 is [ 2 − 1 ] + = 1 [2-1]_+=1 [ 2 − 1 ] + = 1 [ 1 − 1 ] + = 0 [1-1]_+=0 [ 1 − 1 ] + = 0 q v q_v q v
q v ( 1 ) [ 2 − 1 ] + = q v ( 2 ) [ 1 − 1 ] + . q_v(1)[2-1]_+=q_v(2)[1-1]_+. q v ( 1 ) [ 2 − 1 ] + = q v ( 2 ) [ 1 − 1 ] + . which forces q v ( 1 ) = 0 q_v(1)=0 q v ( 1 ) = 0 § 12 .5.4, we consider the distribution
p ~ v ∣ i ( z ) : = ∏ ℓ = 1 L q v ∣ 0 ℓ ( z ℓ ∣ x ^ 0 θ , ℓ ( x , t i ) ) , \widetilde{p}_{v|i}(z):=\prod_{\ell=1}^{L}q^\ell_{v|0}\left(z^\ell\mid\hat{x}^{\theta,\ell}_0(x,t_i)\right), p v ∣ i ( z ) := ℓ = 1 ∏ L q v ∣0 ℓ ( z ℓ ∣ x ^ 0 θ , ℓ ( x , t i ) ) , the product of the forward marginals initialized by the cached leave-one-out predictions. For discrete states z , w z,w z , w
p ~ v ∣ i ( z ) Q ‾ v θ , l o o ( w ; t i ∣ z ) = p ~ v ∣ i ( w ) Q v ( z ∣ w ) , \widetilde{p}_{v|i}(z)\overline{Q}^{\theta,\mathrm{loo}}_v(w;t_i|z)=\widetilde{p}_{v|i}(w)Q_v(z|w), p v ∣ i ( z ) Q v θ , loo ( w ; t i ∣ z ) = p v ∣ i ( w ) Q v ( z ∣ w ) , p ~ v ∣ i ( z ) [ Q ‾ v θ , l o o ( w ; t i ∣ z ) − Q v ( w ∣ z ) ] + = [ p ~ v ∣ i ( w ) Q v ( z ∣ w ) − p ~ v ∣ i ( z ) Q v ( w ∣ z ) ] + = p ~ v ∣ i ( w ) [ Q v ( z ∣ w ) − Q ‾ v θ , l o o ( z ; t i ∣ w ) ] + . \begin{aligned}
\widetilde{p}_{v|i}(z)\left[\overline{Q}^{\theta,\mathrm{loo}}_v(w;t_i|z)-Q_v(w|z)\right]_+
=
\left[\widetilde{p}_{v|i}(w)Q_v(z|w)-\widetilde{p}_{v|i}(z)Q_v(w|z)\right]_+ \\
&=\widetilde{p}_{v|i}(w)\left[Q_v(z|w)-\overline{Q}^{\theta,\mathrm{loo}}_v(z;t_i|w)\right]_+.
\end{aligned} p v ∣ i ( z ) [ Q v θ , loo ( w ; t i ∣ z ) − Q v ( w ∣ z ) ] + = [ p v ∣ i ( w ) Q v ( z ∣ w ) − p v ∣ i ( z ) Q v ( w ∣ z ) ] + = p v ∣ i ( w ) [ Q v ( z ∣ w ) − Q v θ , loo ( z ; t i ∣ w ) ] + . The second identity is the time-reversal balance for the new rates. The net-flux projection also preserves the difference between the two forward fluxes, so the new forward rates still evolve p ~ v ∣ i \widetilde{p}_{v|i} p v ∣ i § 12 .5.4.
Additional material for distillation
Consistency distillation training
The objective and its motivation are given in § 4 ; the round-based training procedure is summarized below. Write D D D
Algorithm 4 Exponential-residual consistency distillation with roundwise target refresh
Require: Pre-trained teacher parameters θ \theta θ ϕ \phi ϕ δ \delta δ N N N M M M η \eta η ρ \rho ρ D D D
Ensure: Distilled student parameters ϕ \phi ϕ
1: ϕ E M A ← ϕ \phi^{\mathrm{EMA}} \leftarrow\phi ϕ EMA ← ϕ for n = 1 , … , N n = 1,\ldots,N n = 1 , … , N do 3: ϕ ˉ ← ϕ E M A \bar{\phi} \leftarrow\phi^{\mathrm{EMA}} ϕ ˉ ← ϕ EMA Freeze the target for this round 4: for m = 1 , … , M m = 1,\ldots,M m = 1 , … , M do 5: Sample t ∼ Unif ( δ , 1 ) t \sim\operatorname{Unif}(\delta,1) t ∼ Unif ( δ , 1 ) s ← t − δ s \leftarrow t-\delta s ← t − δ X 0 ∼ p 0 X_0 \sim p_0 X 0 ∼ p 0 X t ∼ q t ∣ 0 ( ⋅ ∣ X 0 ) X_t \sim q_{t|0}(\cdot|X_0) X t ∼ q t ∣0 ( ⋅ ∣ X 0 ) E t i j ∼ i . i . d . Exp ( 1 ) E_t^{ij} \mathrel{\overset{\mathrm{i.i.d.}}{\sim}} \operatorname{Exp}(1) E t ij ∼ i.i.d. Exp ( 1 ) X t X_t X t ( X s , E s ) ← Ψ s ∣ t θ ( X t , E t ) (X_s,E_s) \leftarrow\Psi^\theta_{s|t}(X_t,E_t) ( X s , E s ) ← Ψ s ∣ t θ ( X t , E t ) L ^ ← D ( p 0 ∣ t ϕ ( ⋅ ∣ X t , E t ) , sg [ p 0 ∣ s ϕ ˉ ( ⋅ ∣ X s , E s ) ] ) \widehat{\mathcal{L}} \leftarrow D\left(p^\phi_{0|t}(\cdot|X_t,E_t),\operatorname{sg}\left[p^{\bar{\phi}}_{0|s}(\cdot|X_s,E_s)\right]\right) L ← D ( p 0∣ t ϕ ( ⋅ ∣ X t , E t ) , sg [ p 0∣ s ϕ ˉ ( ⋅ ∣ X s , E s ) ] ) ϕ ← ϕ − η ∇ ϕ L ^ \phi\leftarrow\phi-\eta\nabla_\phi\widehat{\mathcal{L}} ϕ ← ϕ − η ∇ ϕ L ϕ E M A ← ρ ϕ E M A + ( 1 − ρ ) ϕ \phi^{\mathrm{EMA}} \leftarrow\rho\phi^{\mathrm{EMA}}+(1-\rho)\phi ϕ EMA ← ρ ϕ EMA + ( 1 − ρ ) ϕ end for 13: end for 14: return ϕ \phi ϕ
The joint-refresh variant also replaces the model that supplies the ERF rollout predictions between rounds. The first round uses the pretrained diffusion model. In every later round, the rollout uses the same frozen EMA copy ϕ ˉ \bar{\phi} ϕ ˉ t t t s s s u u u X u X_u X u u u u − log E u -\log E_u − log E u ( X , E ) (X,E) ( X , E )
Algorithm 5. Exponential-residual consistency distillation with roundwise target-and-teacher refresh
Require: Pre-trained teacher parameters θ \theta θ ϕ \phi ϕ δ \delta δ N N N M M M η \eta η ρ \rho ρ D D D
Ensure: Distilled student parameters ϕ \phi ϕ
ϕ E M A ← ϕ \phi^{\mathrm{EMA}} \leftarrow\phi ϕ EMA ← ϕ
for n = 1 , … , N n = 1,\ldots,N n = 1 , … , N do
ϕ ˉ ← ϕ E M A \bar{\phi} \leftarrow\phi^{\mathrm{EMA}} ϕ ˉ ← ϕ EMA
for m = 1 , … , M m = 1,\ldots,M m = 1 , … , M do
Sample t ∼ Unif ( δ , 1 ) t \sim\operatorname{Unif}(\delta,1) t ∼ Unif ( δ , 1 ) s ← t − δ s \leftarrow t-\delta s ← t − δ
Sample X 0 ∼ p 0 X_0 \sim p_0 X 0 ∼ p 0 X t ∼ q t ∣ 0 ( ⋅ ∣ X 0 ) X_t \sim q_{t|0}(\cdot|X_0) X t ∼ q t ∣0 ( ⋅ ∣ X 0 )
Sample E t ℓ j E_t^{\ell j} E t ℓ j ∼ Exp ( 1 ) \sim\operatorname{Exp}(1) ∼ Exp ( 1 ) X t X_t X t
Choose the teacher grid t = u 0 > u 1 > ⋯ > u J = s t = u_0 > u_1 > \cdots> u_J = s t = u 0 > u 1 > ⋯ > u J = s
for q = 0 , … , J − 1 q = 0,\ldots,J-1 q = 0 , … , J − 1 do
if n = 1 n = 1 n = 1 then
Evaluate the pretrained teacher with network inputs ( X u q , u q ) (X_{u_q},u_q) ( X u q , u q )
else
Evaluate the frozen ERC model ϕ ˉ \bar{\phi} ϕ ˉ ( X u q , u q , − log E u q ) (X_{u_q},u_q,-\log E_{u_q}) ( X u q , u q , − log E u q )
end if
Advance ( X u q , E u q ) (X_{u_q},E_{u_q}) ( X u q , E u q ) ( X u q + 1 , E u q + 1 ) (X_{u_{q+1}},E_{u_{q+1}}) ( X u q + 1 , E u q + 1 )
end for
L ^ ← D ( p 0 ∣ t ϕ ( ⋅ ∣ X t , E t ) , sg [ p 0 ∣ s ϕ ˉ ( ⋅ ∣ X s , E s ) ] ) \widehat{\mathcal{L}} \leftarrow D\left(p_{0|t}^{\phi}(\cdot|X_t,E_t),\operatorname{sg}\left[p_{0|s}^{\bar{\phi}}(\cdot|X_s,E_s)\right]\right) L ← D ( p 0∣ t ϕ ( ⋅ ∣ X t , E t ) , sg [ p 0∣ s ϕ ˉ ( ⋅ ∣ X s , E s ) ] )
ϕ ← ϕ − η ∇ ϕ L ^ \phi\leftarrow\phi-\eta\nabla_{\phi}\widehat{\mathcal{L}} ϕ ← ϕ − η ∇ ϕ L
ϕ E M A ← ρ ϕ E M A + ( 1 − ρ ) ϕ \phi^{\mathrm{EMA}} \leftarrow\rho\phi^{\mathrm{EMA}}+(1-\rho)\phi ϕ EMA ← ρ ϕ EMA + ( 1 − ρ ) ϕ
end for
end for
return ϕ \phi ϕ
For UDM, the ERC network natively predicts a leave-one-out distribution. The two uses of this prediction must be distinguished. During a rollout, the pretrained teacher supplies its usual leave-one-out prediction, while a refreshed trajectory teacher supplies the residual-conditioned leave-one-out prediction of the frozen ERC model; in both cases, this leave-one-out distribution is used directly to construct the UDM ERF rates. In the consistency loss, both the current student prediction at ( X t , E t ) (X_t,E_t) ( X t , E t ) ( X s , E s ) (X_s,E_s) ( X s , E s ) D D D E u E_u E u
x ^ 0 ϕ ( x , u ) ℓ = softmax ( log x ^ 0 ω , l o o ( x , u ) ℓ + log ( 1 + K α u 1 − α u ) x ~ ℓ ) (32) \hat{x}_0^{\phi}(x,u)^\ell= \operatorname{softmax}\left(\log\hat{x}_0^{\omega,\mathrm{loo}}(x,u)^\ell+ \log\left(1+\frac{K\alpha_u}{1-\alpha_u}\right)\tilde{x}^\ell\right) \tag*{(32)} x ^ 0 ϕ ( x , u ) ℓ = softmax ( log x ^ 0 ω , loo ( x , u ) ℓ + log ( 1 + 1 − α u K α u ) x ~ ℓ ) (32) Multi-step sampling
Algorithm 6 Multi-step refresh or bridge sampling with a distilled student
Require: Student parameters ϕ \phi ϕ 1 = t k > t k − 1 > ⋯ > t 1 > t 0 = 0 1 = t_k > t_{k-1} > \cdots> t_1 > t_0 = 0 1 = t k > t k − 1 > ⋯ > t 1 > t 0 = 0 s i : = t i − 1 s_i := t_{i-1} s i := t i − 1
Output: A clean state X ^ 0 \widehat{X}_0 X 0
Sample X t k ∼ p 1 X_{t_k} \sim p_1 X t k ∼ p 1 E t k ℓ j ∼ i . i . d . Exp ( 1 ) E_{t_k}^{\ell_j} \overset{\mathrm{i.i.d.}}{\sim} \operatorname{Exp}(1) E t k ℓ j ∼ i.i.d. Exp ( 1 )
for i = k , k − 1 , … , 1 i = k, k-1, \ldots, 1 i = k , k − 1 , … , 1 do
Sample X ^ 0 ∼ p 0 ∣ t i ϕ ( ⋅ ∣ X t i , E t i ) \widehat{X}_0 \sim p_{0|t_i}^{\phi}(\cdot|X_{t_i}, E_{t_i}) X 0 ∼ p 0∣ t i ϕ ( ⋅ ∣ X t i , E t i )
if i > 1 i > 1 i > 1 then
either sample X s i ∼ q s i ∣ 0 ( ⋅ ∣ X ^ 0 ) X_{s_i} \sim q_{s_i|0}(\cdot|\widehat{X}_0) X s i ∼ q s i ∣0 ( ⋅ ∣ X 0 )
or sample X s i ∼ q s i ∣ 0 , t i ( ⋅ ∣ X ^ 0 , X t i ) X_{s_i} \sim q_{s_i|0,t_i}(\cdot|\widehat{X}_0, X_{t_i}) X s i ∼ q s i ∣0 , t i ( ⋅ ∣ X 0 , X t i )
Sample a fresh E s i ℓ j ∼ i . i . d . Exp ( 1 ) E_{s_i}^{\ell_j} \overset{\mathrm{i.i.d.}}{\sim} \operatorname{Exp}(1) E s i ℓ j ∼ i.i.d. Exp ( 1 )
end if
end for
return X ^ 0 \widehat{X}_0 X 0
Bridge sampling and the ERF coupling
The bridge q s ∣ 0 , t ( ⋅ ∣ y , x ) q_{s|0,t}(\cdot|y,x) q s ∣0 , t ( ⋅ ∣ y , x ) (17) . Therefore, to obtain samples from the marginal p s p_s p s X 0 ∼ p 0 ∣ t ( ⋅ ∣ x ) X_0 \sim p_{0|t}(\cdot|x) X 0 ∼ p 0∣ t ( ⋅ ∣ x ) p 0 ∣ t ( ⋅ ∣ x ) p_{0|t}(\cdot|x) p 0∣ t ( ⋅ ∣ x ) p 0 ( y ) q t ∣ 0 ( x ∣ y ) p t ( x ) \frac{p_0(y)q_{t|0}(x|y)}{p_t(x)} p t ( x ) p 0 ( y ) q t ∣0 ( x ∣ y ) X t X_t X t X 0 X_0 X 0
p 0 ∣ t ϕ ( y ∣ x t ) : = E E t [ p 0 ∣ t ϕ ( y ∣ x t , E t ) ] . (33) p_{0|t}^{\phi}(y|x_t) := \mathbb{E}_{E_t}\left[p_{0|t}^{\phi}(y|x_t,E_t)\right]. \tag*{(33)} p 0∣ t ϕ ( y ∣ x t ) := E E t [ p 0∣ t ϕ ( y ∣ x t , E t ) ] . (33) Consequently, the ideal endpoint distribution p 0 ∣ t ϕ ( ⋅ ∣ x t , E t ) p_{0|t}^{\phi}(\cdot|x_t,E_t) p 0∣ t ϕ ( ⋅ ∣ x t , E t ) p 0 ∣ t ( ⋅ ∣ x ) p_{0|t}(\cdot|x) p 0∣ t ( ⋅ ∣ x ) p s p_s p s X ^ 0 ∼ p 0 \widehat{X}_0 \sim p_0 X 0 ∼ p 0 ∑ y p 0 ( y ) q s ∣ 0 ( z ∣ y ) = p s ( z ) \sum_y p_0(y)q_{s|0}(z|y) = p_s(z) ∑ y p 0 ( y ) q s ∣0 ( z ∣ y ) = p s ( z )
For masked diffusion, however, each position jumps at most once, from m to a token. A passive residual changed by ERF can therefore never affect a later jump. The ERF discrete path is therefore equivalent to Gillespie since the passive residuals are never re-used. MDM ERF thus has the original reverse-chain law, so its ideal endpoint student satisfies p 0 ∣ t ϕ ( y ∣ x t ) = p 0 ∣ t ( y ∣ x t ) p_{0|t}^{\phi}(y|x_t) = p_{0|t}(y|x_t) p 0∣ t ϕ ( y ∣ x t ) = p 0∣ t ( y ∣ x t )
D-MMD
We implement D-MMD following Hoogeboom et al. [20 ] . Let ϕ \phi ϕ θ \theta θ ψ \psi ψ
X 0 ∼ p 0 , X t ∼ q t ∣ 0 ( ⋅ ∣ X 0 ) , X ^ 0 ℓ ∼ Cat ( α ^ 0 ϕ ( X t , t ) ℓ ) , X s ∼ q s ∣ 0 , t ( ⋅ ∣ X ^ 0 , X t ) , s < t . X_0 \sim p_0,\quad X_t \sim q_{t|0}(\cdot|X_0),\quad\widehat{X}_0^\ell\sim\operatorname{Cat}\left(\widehat{\alpha}^{\phi}_0(X_t,t)^\ell\right),\quad X_s \sim q_{s|0,t}(\cdot|\widehat{X}_0,X_t),\quad s < t. X 0 ∼ p 0 , X t ∼ q t ∣0 ( ⋅ ∣ X 0 ) , X 0 ℓ ∼ Cat ( α 0 ϕ ( X t , t ) ℓ ) , X s ∼ q s ∣0 , t ( ⋅ ∣ X 0 , X t ) , s < t . All sampled states, including X ^ 0 \widehat{X}_0 X 0 X s X_s X s [20 ] . During the generator update, only the explicit student probabilities are differentiated; teacher and auxiliary predictions are fixed. During the auxiliary update, the student and teacher predictions are fixed and only the auxiliary parameters are differentiated. Thus the displayed expressions specify surrogate losses with these stop-gradient conventions, rather than derivatives through the parameter-dependent sampling law. Writing w s = ( 1 − α s ) − 1 w_s = (1-\alpha_s)^{-1} w s = ( 1 − α s ) − 1
L g e n ( ϕ ) = − E [ w s ∑ ℓ : X s ℓ = m ⟨ x ^ 0 ϕ ( X t , t ) ℓ , log x ^ 0 θ ( X s , s ) ℓ − log x ^ 0 ψ ( X s , s ) ℓ ⟩ ] , L a u x ( ψ ) = − E [ w s ∑ ℓ : X s ℓ = m ⟨ x ^ 0 ϕ ( X t , t ) ℓ + x ^ 0 θ ( X s , s ) ℓ , log x ^ 0 ψ ( X s , s ) ℓ ⟩ ] . (34) \begin{aligned}
\mathcal{L}_{\mathrm{gen}}(\phi) &= -\mathbb{E}\left[w_s \sum_{\ell:X_s^\ell=m}\left\langle\hat{x}_0^\phi(X_t,t)^\ell,\log\hat{x}_0^\theta(X_s,s)^\ell-\log\hat{x}_0^\psi(X_s,s)^\ell\right\rangle\right],\\
\mathcal{L}_{\mathrm{aux}}(\psi) &= -\mathbb{E}\left[w_s \sum_{\ell:X_s^\ell=m}\left\langle\hat{x}_0^\phi(X_t,t)^\ell+\hat{x}_0^\theta(X_s,s)^\ell,\log\hat{x}_0^\psi(X_s,s)^\ell\right\rangle\right]. \tag*{(34)}
\end{aligned} L gen ( ϕ ) L aux ( ψ ) = − E w s ℓ : X s ℓ = m ∑ ⟨ x ^ 0 ϕ ( X t , t ) ℓ , log x ^ 0 θ ( X s , s ) ℓ − log x ^ 0 ψ ( X s , s ) ℓ ⟩ , = − E w s ℓ : X s ℓ = m ∑ ⟨ x ^ 0 ϕ ( X t , t ) ℓ + x ^ 0 θ ( X s , s ) ℓ , log x ^ 0 ψ ( X s , s ) ℓ ⟩ . (34) For uniform diffusion, the student, teacher, and auxiliary model all use leave-one-out predictions. We sample X ~ 0 ℓ ∼ Cat ( x ^ 0 ϕ , l o o ( X t , t ) ℓ ) \widetilde{X}_0^\ell\sim\operatorname{Cat}(\hat{x}_0^{\phi,\mathrm{loo}}(X_t,t)^\ell) X 0 ℓ ∼ Cat ( x ^ 0 ϕ , loo ( X t , t ) ℓ ) X s ∼ q s ∣ 0 , t ( ⋅ ∣ X ~ 0 , X t ) X_s \sim q_{s|0,t}(\cdot|\widetilde{X}_0,X_t) X s ∼ q s ∣0 , t ( ⋅ ∣ X 0 , X t ) [20 ] : for r < s r<s r < s
L g e n l o o ( ϕ ) = E [ 1 s − r ∑ ℓ = 1 L ⟨ q r ∣ 0 , s ℓ ( ⋅ ∣ x ^ 0 ϕ , l o o ( X t , t ) ℓ , X s ℓ ) , log q r ∣ 0 , s ℓ ( ⋅ ∣ x ^ 0 ψ , l o o ( X s , s ) ℓ , X s ℓ ) − log q r ∣ 0 , s ℓ ( ⋅ ∣ x ^ 0 θ , l o o ( X s , s ) ℓ , X s ℓ ) ⟩ ] , \mathcal{L}_{\mathrm{gen}}^{\mathrm{loo}}(\phi)=\mathbb{E}\left[\frac{1}{s-r}\sum_{\ell=1}^{L}\left\langle q_{r|0,s}^{\ell}(\cdot|\hat{x}_0^{\phi,\mathrm{loo}}(X_t,t)^\ell,X_s^\ell),\log q_{r|0,s}^{\ell}(\cdot|\hat{x}_0^{\psi,\mathrm{loo}}(X_s,s)^\ell,X_s^\ell)-\log q_{r|0,s}^{\ell}(\cdot|\hat{x}_0^{\theta,\mathrm{loo}}(X_s,s)^\ell,X_s^\ell)\right\rangle\right], L gen loo ( ϕ ) = E [ s − r 1 ℓ = 1 ∑ L ⟨ q r ∣0 , s ℓ ( ⋅ ∣ x ^ 0 ϕ , loo ( X t , t ) ℓ , X s ℓ ) , log q r ∣0 , s ℓ ( ⋅ ∣ x ^ 0 ψ , loo ( X s , s ) ℓ , X s ℓ ) − log q r ∣0 , s ℓ ( ⋅ ∣ x ^ 0 θ , loo ( X s , s ) ℓ , X s ℓ ) ⟩ ] , L a u x l o o ( ψ ) = − E [ 1 s − r ∑ ℓ = 1 L ⟨ q r ∣ 0 , s ℓ ( ⋅ ∣ X ~ 0 ℓ , X s ℓ ) + q r ∣ 0 , s ℓ ( ⋅ ∣ x ^ 0 θ , l o o ( X s , s ) ℓ , X s ℓ ) , log q r ∣ 0 , s ℓ ( ⋅ ∣ x ^ 0 ψ , l o o ( X s , s ) ℓ , X s ℓ ) ⟩ ] . \mathcal{L}_{\mathrm{aux}}^{\mathrm{loo}}(\psi)=-\mathbb{E}\left[\frac{1}{s-r}\sum_{\ell=1}^{L}\left\langle q_{r|0,s}^{\ell}(\cdot|\widetilde{X}_0^\ell,X_s^\ell)+q_{r|0,s}^{\ell}(\cdot|\hat{x}_0^{\theta,\mathrm{loo}}(X_s,s)^\ell,X_s^\ell),\log q_{r|0,s}^{\ell}(\cdot|\hat{x}_0^{\psi,\mathrm{loo}}(X_s,s)^\ell,X_s^\ell)\right\rangle\right]. L aux loo ( ψ ) = − E [ s − r 1 ℓ = 1 ∑ L ⟨ q r ∣0 , s ℓ ( ⋅ ∣ X 0 ℓ , X s ℓ ) + q r ∣0 , s ℓ ( ⋅ ∣ x ^ 0 θ , loo ( X s , s ) ℓ , X s ℓ ) , log q r ∣0 , s ℓ ( ⋅ ∣ x ^ 0 ψ , loo ( X s , s ) ℓ , X s ℓ ) ⟩ ] . DCD
For a fair comparison, every uniform-diffusion baseline starts from the same checkpoints trained with the UDM objective. We therefore implement DCD from Sahoo et al. [27 ] directly in the UDM state space by marginalizing its Gaussian construction. Sahoo et al. draw G ℓ j G^{\ell j} G ℓ j ∼ N ( 0 , 1 ) \sim\mathcal{N}(0,1) ∼ N ( 0 , 1 )
X u ℓ = argmax j ∈ [ 1 , K ] { α ~ u 1 j = X 0 ℓ + 1 − α ~ u 2 G ℓ j } , u ∈ { s , t } . X_u^\ell=\underset{j\in[1,K]}{\operatorname{argmax}}\left\{\widetilde{\alpha}_u\mathbf{1}_{j=X_0^\ell}+\sqrt{1-\widetilde{\alpha}_u^2}G^{\ell j}\right\},\qquad u\in\{s,t\}. X u ℓ = j ∈ [ 1 , K ] argmax { α u 1 j = X 0 ℓ + 1 − α u 2 G ℓ j } , u ∈ { s , t } . The same G ℓ j G^{\ell j} G ℓ j α ~ u \widetilde{\alpha}_u α u X u ℓ ∣ X 0 ℓ ∼ Cat ( α u X ~ 0 ℓ + ( 1 − α u ) 1 / K ) X_u^\ell\mid X_0^\ell\sim\operatorname{Cat}(\alpha_u\widetilde{X}_0^\ell+(1-\alpha_u)\mathbf{1}/K) X u ℓ ∣ X 0 ℓ ∼ Cat ( α u X 0 ℓ + ( 1 − α u ) 1 / K ) G ℓ j G^{\ell j} G ℓ j j ≠ X 0 ℓ j\ne X_0^\ell j = X 0 ℓ α ~ s / 1 − α ~ s 2 ≥ α ~ t / 1 − α ~ t 2 \widetilde{\alpha}_s/\sqrt{1-\widetilde{\alpha}_s^2}\ge\widetilde{\alpha}_t/\sqrt{1-\widetilde{\alpha}_t^2} α s / 1 − α s 2 ≥ α t / 1 − α t 2 X t ℓ = X 0 ℓ X_t^\ell=X_0^\ell X t ℓ = X 0 ℓ X s ℓ = X 0 ℓ X_s^\ell=X_0^\ell X s ℓ = X 0 ℓ X t ℓ ≠ X 0 ℓ X_t^\ell\ne X_0^\ell X t ℓ = X 0 ℓ X s ℓ ∈ { X t ℓ , X 0 ℓ } X_s^\ell\in\{X_t^\ell,X_0^\ell\} X s ℓ ∈ { X t ℓ , X 0 ℓ } { X s ℓ = j } ⊆ { X t ℓ = j } \{X_s^\ell=j\}\subseteq\{X_t^\ell=j\} { X s ℓ = j } ⊆ { X t ℓ = j } j ≠ X 0 ℓ j\ne X_0^\ell j = X 0 ℓ
P ( X s ℓ = j ∣ X t ℓ = j , X 0 ℓ ) = P ( X s ℓ = j ∣ X 0 ℓ ) P ( X t ℓ = j ∣ X 0 ℓ ) = 1 − α s 1 − α t . \mathbb{P}(X_s^\ell=j\mid X_t^\ell=j,X_0^\ell)=\frac{\mathbb{P}(X_s^\ell=j\mid X_0^\ell)}{\mathbb{P}(X_t^\ell=j\mid X_0^\ell)}=\frac{1-\alpha_s}{1-\alpha_t}. P ( X s ℓ = j ∣ X t ℓ = j , X 0 ℓ ) = P ( X t ℓ = j ∣ X 0 ℓ ) P ( X s ℓ = j ∣ X 0 ℓ ) = 1 − α t 1 − α s . The remaining probability goes to X 0 ℓ X_0^\ell X 0 ℓ
X s ℓ ∣ ( X 0 ℓ , X t ℓ ) ∼ Cat ( α s − α t 1 − α t x ~ 0 ℓ + 1 − α s 1 − α t x ~ t ℓ ) . X_s^\ell\mid(X_0^\ell,X_t^\ell)\sim\operatorname{Cat}\left(\frac{\alpha_s-\alpha_t}{1-\alpha_t}\widetilde{x}_0^\ell+\frac{1-\alpha_s}{1-\alpha_t}\widetilde{x}_t^\ell\right). X s ℓ ∣ ( X 0 ℓ , X t ℓ ) ∼ Cat ( 1 − α t α s − α t x 0 ℓ + 1 − α t 1 − α s x t ℓ ) . This is a maximal coupling of the two one-position UDM marginals conditional on X 0 ℓ X_0^\ell X 0 ℓ [18 ] ). Therefore, this transition can be reproduced without the underlying Gaussian trajectory. We sample it directly, without Gaussian variables, by drawing independently across positions U ℓ ∼ Unif ( 0 , 1 ) U^\ell\sim\operatorname{Unif}(0,1) U ℓ ∼ Unif ( 0 , 1 ) Y ℓ ∼ Cat ( 1 / K ) Y^\ell\sim\operatorname{Cat}(1/K) Y ℓ ∼ Cat ( 1/ K )
X u ℓ = { Y ℓ , U ℓ < 1 − α u , X 0 ℓ , U ℓ ≥ 1 − α u , u ∈ { s , t } . X_u^\ell=
\begin{cases}
Y^\ell, & U^\ell<1-\alpha_u,\\
X_0^\ell, & U^\ell\ge1-\alpha_u,
\end{cases}
\qquad u\in\{s,t\}. X u ℓ = { Y ℓ , X 0 ℓ , U ℓ < 1 − α u , U ℓ ≥ 1 − α u , u ∈ { s , t } . The coupling induces the transition
p s ∣ t max ( x s ∣ x t ) = ∑ x 0 ∈ X p 0 ∣ t ( x 0 ∣ x t ) ∏ ℓ = 1 L Cat ( x s ℓ ; α s − α t 1 − α t x ~ 0 ℓ + 1 − α s 1 − α t x ~ t ℓ ) . p_{s|t}^{\max}(x_s|x_t)=\sum_{x_0\in\mathcal{X}}p_{0|t}(x_0|x_t)\prod_{\ell=1}^{L}\operatorname{Cat}\left(x_s^\ell;\frac{\alpha_s-\alpha_t}{1-\alpha_t}\widetilde{x}_0^\ell+\frac{1-\alpha_s}{1-\alpha_t}\widetilde{x}_t^\ell\right). p s ∣ t m a x ( x s ∣ x t ) = x 0 ∈ X ∑ p 0∣ t ( x 0 ∣ x t ) ℓ = 1 ∏ L Cat ( x s ℓ ; 1 − α t α s − α t x 0 ℓ + 1 − α t 1 − α s x t ℓ ) . At the beginning of each round, we freeze the current student parameters ϕ − \phi^{-} ϕ −
ϕ − ← ϕ , s = ( t − Δ k ) + , Δ k + 1 = 2 Δ k . \phi^{-} \leftarrow\phi,\qquad s=(t-\Delta_k)_{+},\qquad\Delta_{k+1}=2\Delta_k. ϕ − ← ϕ , s = ( t − Δ k ) + , Δ k + 1 = 2 Δ k . We convert both leave-one-out outputs to mean predictions using the UDM identity above and minimize the reverse KL
L D C D ( ϕ ; ϕ − ) = E [ 1 L ∑ ℓ = 1 L K L ( x ~ 0 ϕ ( X t , t ) ℓ ∥ x ~ 0 ϕ − ( X s , s ) ℓ ) ] . (35) \mathcal{L}_{\mathrm{DCD}}(\phi;\phi^{-})=\mathbb{E}\left[\frac{1}{L}\sum_{\ell=1}^{L}\mathrm{KL}\left(\widetilde{x}_{0}^{\phi}(X_t,t)^{\ell}\mathbin{\|}\widetilde{x}_{0}^{\phi^{-}}(X_s,s)^{\ell}\right)\right]. \tag*{(35)} L DCD ( ϕ ; ϕ − ) = E [ L 1 ℓ = 1 ∑ L KL ( x 0 ϕ ( X t , t ) ℓ ∥ x 0 ϕ − ( X s , s ) ℓ ) ] . (35) For fixed ϕ − \phi^{-} ϕ − t t t (35) is therefore, for j ∈ ⟦ 1 , K ⟧ j\in\llbracket1,K\rrbracket j ∈ [ [ 1 , K ] ]
⟨ j ~ , x ^ 0 ϕ ∗ ( x t , t ) ℓ ⟩ = exp { E X s ∼ p s ∣ t max ( ⋅ ∣ x t ) [ log k ~ , x ^ 0 ϕ − ( X s , s ) ℓ ] } ∑ k = 1 K exp { E X s ∼ p s ∣ t max ( ⋅ ∣ x t ) [ log k ~ , x ^ 0 ϕ − ( X s , s ) ℓ ] } . \left\langle\widetilde{j},\widehat{x}_{0}^{\phi^{*}}(x_t,t)^{\ell}\right\rangle
=
\frac{\exp\left\{\mathbb{E}_{X_s\sim p_{s\mid t}^{\max}(\cdot\mid x_t)}\left[\log\widetilde{k},\widehat{x}_{0}^{\phi^{-}}(X_s,s)^{\ell}\right]\right\}}
{\sum_{k=1}^{K}\exp\left\{\mathbb{E}_{X_s\sim p_{s\mid t}^{\max}(\cdot\mid x_t)}\left[\log\widetilde{k},\widehat{x}_{0}^{\phi^{-}}(X_s,s)^{\ell}\right]\right\}}. ⟨ j , x 0 ϕ ∗ ( x t , t ) ℓ ⟩ = ∑ k = 1 K exp { E X s ∼ p s ∣ t m a x ( ⋅ ∣ x t ) [ log k , x 0 ϕ − ( X s , s ) ℓ ] } exp { E X s ∼ p s ∣ t m a x ( ⋅ ∣ x t ) [ log k , x 0 ϕ − ( X s , s ) ℓ ] } . If a target probability can vanish, this expression instead requires a nonempty common support with a positive finite normalizing constant. In particular, the exact mean-prediction boundary at s = 0 s=0 s = 0 X 0 ℓ X_{0}^{\ell} X 0 ℓ X t X_t X t s = 0 s=0 s = 0 α s = 1 \alpha_s=1 α s = 1 X s ℓ ∼ q s ∣ 0 ϕ ( ⋅ ∣ x ^ 0 ϕ ( X t , t ) ℓ ) X_s^{\ell}\sim q_{s\mid0}^{\phi}(\cdot\mid\widehat{x}_{0}^{\phi}(X_t,t)^{\ell}) X s ℓ ∼ q s ∣ 0 ϕ ( ⋅ ∣ x 0 ϕ ( X t , t ) ℓ )
iDLM
We implement iDLM ([21 ] ) with a frozen teacher, a student, and a trainable auxiliary denoiser, initialized from the same pretrained checkpoint used by the other methods in the corresponding diffusion family. Training alternates student and auxiliary updates. The student minimizes the difference between teacher and auxiliary denoising losses on its generated distribution, while the auxiliary denoiser is trained on that distribution. For masked diffusion, sampled categorical states are held fixed during differentiation and gradients act on the student probabilities appearing explicitly in the loss.
For uniform diffusion, we retain the Gaussian relaxation used by iDLM to obtain differentiable noisy inputs, with relaxation temperature 10 − 3 10^{-3} 1 0 − 3 § 14 .§ 5 . At evaluation, iDLM generates discrete tokens and is subject to the same sampler and temperature controls as the other students.
Experimental details
Sudoku
We train one MDM and one UDM teacher per difficulty. Each example concatenates an 81-digit quiz, a separator, its 81-digit solution, and an end-of-sequence token; 0 denotes a blank quiz position. The quiz and separator remain fixed, while the model predicts the solution positions. The vocabulary includes ten digits and five special tokens for padding, separation, masking, end of sequence, and unknown inputs. Table 3 to Table 5 summarize the configuration. Exact-grid accuracy requires all 81 predicted positions to match the reference solution; constraint violations are also computed.
Table 3. Sudoku datasets, generated with seed 42.
Table 5. Shared Sudoku teacher training recipe, applied to all six runs.
Table 4. Sudoku teacher architecture and diffusion configuration.
Additional experiments
Sampling quality across NFE budgets
Figures 5 and 6 extend the main comparison to all eight evaluated NFE budgets, from 8 to 1,024. The samplers share a pretrained checkpoint within each diffusion family. TinyGSM evaluations use all 1,319 GSM8K test questions and temperatures { 0.1 , 0.2 , … , 1 } \{0.1, 0.2, \ldots, 1\} { 0.1 , 0.2 , … , 1 } { 0.80 , 0.82 , … , 1.10 } \{0.80, 0.82, \ldots, 1.10\} { 0.80 , 0.82 , … , 1.10 }
Figure 5. TinyGSM sampling across all evaluated NFE budgets. Each panel gives GSM8K accuracy across sampling temperatures, with pointwise 95% bootstrap intervals. Dashed lines denote ancestral sampling and solid lines denote ERF.
Figure 6. OpenWebText sampling across all evaluated NFE budgets. Markers follow the temperature grid from 0.80 to 1.10, with stars at 1. Perplexity uses an inverted logarithmic axis, scaled independently in each panel.
Euler and categorical tau-leaping
Figure 7 to Figure 9 compare ERF with the Euler and categorical τ \tau τ § 12 .§ 4 , separately for masked, uniform, and U-NetFlux rates at 8, 64, and 512 NFEs. Each rate family uses one pretrained checkpoint, and the masked and uniform comparisons also include the ancestral controls from the main sampling experiment. TinyGSM evaluations use all 1,319 GSM8K test questions at temperatures { 0.1 , 0.2 , … , 1 } \{0.1, 0.2, \ldots, 1\} { 0.1 , 0.2 , … , 1 } { 0.80 , 0.82 , … , 1.10 } \{0.80, 0.82, \ldots, 1.10\} { 0.80 , 0.82 , … , 1.10 } τ \tau τ
Figure 7. ERF, ancestral sampling, Euler, and categorical τ \tau τ
Figure 9. ERF U-NetFlux, Euler, and categorical τ \tau τ Figure 7 . Each panel compares the three integration schemes with the same pretrained uniform-diffusion checkpoint.
The Euler implementation clips the categorical weights and normalizes them when the frozen-rate step violates the probability constraint, while categorical τ \tau τ
Figure 8. ERF Uniform, ancestral UDM, Euler, and categorical τ \tau τ Figure 7 . The ancestral reference uses temperature scaling before prediction conversion and 960 OpenWebText sequences per point; the other methods use scaling after conversion and 480 sequences. All four samplers share the same pretrained checkpoint.
Uniform substeps
The bounded variants divide each frozen-model interval into 1, 2, 4, 8, 16, or 32 substeps and permit at most one jump per position within a substep. The unbounded variant resolves every crossing. Substeps do not introduce additional model evaluations. Figure 10 and Figure 11 display all completed settings across the eight NFE budgets. The TinyGSM grid uses temperatures 0.1, 0.5, and 1; the OpenWebText grid uses { 0.80 , 0.84 , … , 1.08 } \{0.80, 0.84, \ldots, 1.08\} { 0.80 , 0.84 , … , 1.08 }
Runtime and jump-count protocol
The profiles use one AMD MI210, batches of four sequences, and generator seeds 1, 2, and 3. For TinyGSM, the three batches contain the first twelve GSM8K test prompts, four successive prompts per seed. Prompt positions remain fixed in a 512-token canvas. OpenWebText generation is unconditional with length 1,024. Sampling temperatures are 0.1 for TinyGSM and 1 for OpenWebText. Both methods use the same checkpoint, bfloat16 model autocast, and float64 residual clocks.
Uniform explicitly uses 40 bisection iterations per crossing-time solve. Its terminal time is 10 − 6 10^{-6} 1 0 − 6
Jump counts come from a separate replay with the same generator seed and prompt batch. We check that its final tokens agree exactly with the uninstrumented run. Counts measure actual token changes and are divided by the number of generated positions, excluding fixed prompt positions. Figure 12 exposes the three individual measurements; Table 6 gives their means. These runs measure sampling cost on the stated batches.
Figure 12. Individual runtime and jump-count measurements. Faint curves and points show the three seeds; the main curves show their means. In the right column, the main curve is the ratio of mean Uniform and U-NetFlux runtimes, while faint curves give the three paired ratios. The dotted line marks equal runtime.
Table 6. Mean sampling time in seconds per sequence, speedup, and jumps per generated position across the three profile batches. Speedup is the ratio of mean Uniform and U-NetFlux runtimes.
Distillation evaluation
The main distillation figures use the completed runs with budgets of 50 GPU-hours per student on TinyGSM and 100 GPU-hours on OpenWebText, all with global batch size 128. The budget is the number of allocated GPUs multiplied by elapsed training-job time, including initialization, checkpointing and training validation, but excluding queue time and independent evaluation jobs. The number of student updates therefore varies by method. All methods share the original temperature-1 teacher checkpoint within each dataset and diffusion family, with top-p p p § 14 .§ 6 .
The main figures use the final student EMA checkpoints from training seed 1. All student and matching original-teacher evaluations use sampling seed 101, so no across-seed uncertainty band is shown. ERC refreshes its frozen EMA target and rollout teacher at 25%, 50% and 75% of the training budget. We evaluate gaps δ ∈ { 1 / 8 , 1 / 16 , 1 / 32 , 1 / 64 } \delta\in\{1/8, 1/16, 1/32, 1/64\} δ ∈ { 1/8 , 1/16 , 1/32 , 1/64 }
The main DCD curves use the final native consistency-distillation runs. The completed evaluation grid contains both bridge and refresh sampling for every student and teacher, with denoiser temperature placement for MDM and both denoiser and LOO placement for UDM. At temperature 1, the two UDM temperature placements reuse the same evaluation within each sampler; these entries are aliases, not independent samples.
TinyGSM students and teachers in the main figure are evaluated on the same first 1,024 GSM8K test questions, with a 512-token canvas. The common sampling temperatures are 0.1, 0.23, 0.36, 0.49, 0.61, 0.74, 0.87 and 1, evaluated at { 2 , 4 , 8 , 16 , 32 , 64 } \{2, 4, 8, 16, 32, 64\} { 2 , 4 , 8 , 16 , 32 , 64 } § 16 .9.
The main OpenWebText figure uses bridge sampling and denoiser temperature placement for every method and teacher, at temperatures 0.8, 0.85, 0.9, 0.95, 1, 1.05, 1.1 and 1.2. Every point uses 2,048 sequences of length 1,024. To select the ERC gap within each family, we restrict each NFE comparison to the entropy interval shared by all four gaps, linearly interpolate log-perplexity as a function of entropy, and integrate over that interval, dividing by its width. We choose the gap with the lowest average of these normalized integrals over { 2 , 4 , 8 , 16 , 32 , 64 } \{2, 4, 8, 16, 32, 64\} { 2 , 4 , 8 , 16 , 32 , 64 } § 16 .8 and § 16 .9 give the complete measured gap and sampler sweeps; their OpenWebText panels fit the visible points subject to entropy bounds 4–6 and a generative-perplexity ceiling of 1,000, clipping values outside these limits.
Sudoku distillation
Tables 7 and 8 report exact-grid accuracy on the 2,000 validation puzzles at each difficulty and NFE budget in { 1 , 2 , 4 , 8 , 16 , 32 , 64 } \{1, 2, 4, 8, 16, 32, 64\} { 1 , 2 , 4 , 8 , 16 , 32 , 64 } { 0.1 , 0.25 , 0.5 , 0.7 , 0.8 , 0.9 , 1 , 1.1 } \{0.1, 0.25, 0.5, 0.7, 0.8, 0.9, 1, 1.1\} { 0.1 , 0.25 , 0.5 , 0.7 , 0.8 , 0.9 , 1 , 1.1 }
Table 8. Sudoku UDM exact-grid accuracy in percent. Mean- and LOO-parameterized UDM use distinct teachers and method coverage. Selection, subscripts and boldface follow Table 7 .
Table 7. Sudoku MDM exact-grid accuracy in percent. Subscripts report the selected sampler and temperature, with R denoting categorical refresh and B denoting the ancestral bridge. Bold denotes the highest accuracy within each difficulty and NFE budget.
Target and teacher updates
The target is the stopped-gradient student on the right of the consistency loss; the transport teacher is the denoiser used to generate the paired ERF states. At a refresh, the target is replaced by the student EMA; target-and-teacher refresh also uses that copy for transport. Over 10,000 updates, we compare a frozen target, per-step EMA, target refresh every 2,500 updates, and target-and-teacher refresh. Each uses either a fixed gap δ = 1 / 64 \delta= 1/64 δ = 1/64 1 / 64 1/64 1/64 1 / 8 1/8 1/8
Figure 13. TinyGSM target-update policies with a fixed training gap, under bridge sampling.
Figure 14. The same fixed-gap policies under refresh sampling.
Figure 15. TinyGSM target-update policies with an increasing training gap, under bridge sampling.
Figure 16. The same increasing-gap policies under refresh sampling.
Figure 17. OWT target-update policies with a fixed training gap, under bridge sampling. Stars mark temperature 1.
Figure 18. The same OWT policies with an increasing gap. Axes fit visible points within entropy 4–6 and perplexity at most 1,000.
Consistency gap
We compare δ ∈ { 1 / 8 , 1 / 16 , 1 / 32 , 1 / 64 } \delta\in\{1/8, 1/16, 1/32, 1/64\} δ ∈ { 1/8 , 1/16 , 1/32 , 1/64 } Figure 19 uses one sampler and temperature selected by mean accuracy across the six NFEs. For OWT, Figure 20 averages log-perplexity over the entropy interval shared by all eight curves at each NFE, then across NFEs; lower scores correspond to lower matched-entropy perplexity. The measured OWT temperature curves appear in Figure 21 to Figure 24 .
Figure 19. TinyGSM accuracy across training gaps and refresh policies. Solid curves refresh only the target; dashed curves also refresh the rollout teacher.
Figure 20. OWT matched-entropy generative perplexity, averaged across six NFEs, for each training gap and refresh policy.
Figure 21. OWT MDM with target refresh: measured bridge-sampling temperature curves at each NFE. Stars mark temperature 1.
Figure 24. OWT UDM with joint target-and-teacher refresh, under the same evaluation grid as Figure 23 .
Figure 23. OWT UDM with target refresh. Axes show visible points within entropy 4–6 and perplexity at most 1,000.
Figure 22. OWT MDM with joint target-and-teacher refresh, under the same evaluation grid as Figure 21 .
Bridge and refresh sampling
We compare bridge sampling, which conditions the next state on the current state and the student’s clean prediction, with refresh sampling, which re-noises that prediction and draws new residuals. For UDM, temperature is applied either to LOO probabilities before conversion or to denoising probabilities after conversion; MDM uses denoiser temperature. Each comparison includes the shared teacher, iDLM, D-MMD, SDTT for MDM or DCD for UDM, and ERC. ERC uses the selected gap in Appendix § 16 .§ 5 , fixed for each dataset and diffusion family.
For TinyGSM, Figure 25 to Figure 27 show accuracy against NFE with bridge and refresh in adjacent panels. Within each panel, one of the eight measured temperatures is selected separately for each method by its mean accuracy over the six NFE budgets; that temperature is printed in the legend and held fixed across NFEs. Figure 28 to Figure 30 instead show all measured temperatures at every NFE for both samplers, with pointwise 95% question-bootstrap bands. All points use the same 1,024 GSM8K questions.
Figure 25. TinyGSM MDM accuracy versus NFE under bridge and refresh sampling. Each method retains the temperature selected by mean accuracy across six NFE budgets; the teacher, SDTT, iDLM, D-MMD, and ERC are included.
Figure 27. TinyGSM UDM accuracy versus NFE with temperature applied to LOO predictions. Bridge and refresh panels contain the teacher, iDLM, D-MMD, DCD, and ERC.
Figure 28. TinyGSM MDM accuracy across eight denoiser temperatures and six NFE budgets for bridge and refresh sampling. Shaded bands are pointwise 95% bootstrap intervals over questions.
Figure 30. TinyGSM UDM LOO-temperature sweeps for bridge and refresh sampling. All eight temperatures and six NFE budgets are retained, with pointwise 95% question-bootstrap intervals.
For OpenWebText, each panel of Figure 31 to Figure 36 fixes an NFE budget and sweeps the eight measured temperatures. Axes fit the visible points without extending below entropy 4, above entropy 6, or above generative perplexity 1,000; points outside these bounds are not shown, and a star identifies temperature 1 when visible. The perplexity axis is logarithmic and inverted. All points use 2,048 generated sequences. Denoiser and LOO placement at temperature 1 are aliases of one evaluation within each UDM sampler, not independent replicates.
Figure 31. OWT MDM entropy–perplexity curves under bridge sampling.
Figure 36. The same OWT UDM LOO-temperature comparison under refresh sampling.
Figure 26. TinyGSM UDM accuracy versus NFE with temperature applied to denoiser predictions, comparing bridge and refresh sampling and all five methods.
Figure 29. TinyGSM UDM denoiser-temperature sweeps for both samplers at all six NFE budgets. Bands give pointwise 95% question-bootstrap intervals.
Figure 32. The same OWT MDM comparison under refresh sampling.
Figure 33. OWT UDM entropy–perplexity curves with denoiser temperature under bridge sampling.
Figure 34. The same OWT UDM denoiser-temperature comparison under refresh sampling.
Figure 35. OWT UDM entropy–perplexity curves with LOO temperature under bridge sampling.
Gumbel conditioning
We compare ERC with and without conditioning on the Gumbel variables − log E t -\log E_t − log E t
Figure 37. TinyGSM MDM with and without Gumbel inputs under bridge and refresh sampling. Points average two sampling seeds.
Figure 38. TinyGSM UDM under the same Gumbel-input comparison, with an increasing training gap.
Figure 39. OWT Gumbel-input comparison for MDM and UDM under bridge sampling at five NFEs and fourteen temperatures.
DiffusionGemma experiment
We evaluate the instruction-tuned 26B-parameter, 4B-active DiffusionGemma checkpoint (Google DeepMind, 2026) on the MMLU-Pro test split (Wang et al., 2024). We select ten questions from each of its fourteen categories by a stable hash with seed 20260820, yielding the same 140 questions for run seeds 1, 2, and 3. Each prompt requests a concise justification ending with The answer is (X)., and generation uses one canvas block of at most 256 tokens. The strict evaluator accepts only this final-answer form and counts malformed or truncated completions as incorrect.
We compare the official native generation procedure with UDLM and the unbounded ERF Uniform and ERF U-NetFlux samplers. The custom samplers use 48 NFEs, temperature 1, and temperature scaling of the raw denoiser logits. The accuracy intervals in Table 1 use 50,000 category-stratified bootstrap replicates over question clusters while retaining all three paired seeds. The native procedure terminates adaptively after 11.1 forward evaluations on average, whereas each custom sampler uses exactly 48; the reported timings therefore describe the runs but do not constitute a compute-matched comparison. This MMLU-Pro evaluation is not directly comparable to evaluations on other benchmarks reported with the model release.
Proofs for the ERF sampler
Intuition on the ODEs (9)
To motivate the form of this ODE, we derive a continuity equation on the extended state space X × [ 0 , ∞ ) L × K \mathcal{X} \times[0,\infty)^{L \times K} X × [ 0 , ∞ ) L × K X t X_t X t ( Q t ) t ∈ [ 0 , 1 ] (Q_t)_{t \in[0,1]} ( Q t ) t ∈ [ 0 , 1 ] (5) and using (6) , we have for any x x x t t t
∂ t p t ( x ) = p t ( x ) ∑ ℓ = 1 L ( b t ℓ ( x ) − λ t ℓ ( x ℓ ) ) , (36) \partial_t p_t(x) = p_t(x) \sum_{\ell=1}^{L} \left(b_t^\ell(x) - \lambda_t^\ell(x^\ell)\right), \tag*{(36)} ∂ t p t ( x ) = p t ( x ) ℓ = 1 ∑ L ( b t ℓ ( x ) − λ t ℓ ( x ℓ ) ) , (36) where b t ℓ ( x ) : = ∑ j ∈ V ∖ { x ℓ } Q ‾ t ℓ ( j ∣ x ) b_t^\ell(x) := \sum_{j\in\mathcal{V}\setminus\{x^\ell\}} \overline{Q}_t^\ell(j|x) b t ℓ ( x ) := ∑ j ∈ V ∖ { x ℓ } Q t ℓ ( j ∣ x ) ℓ \ell ℓ λ t ℓ ( x ℓ ) : = ∑ j ∈ V ∖ { x ℓ } Q t ℓ ( j ∣ x ℓ ) \lambda_t^\ell(x^\ell) := \sum_{j\in\mathcal{V}\setminus\{x^\ell\}} Q_t^\ell(j|x^\ell) λ t ℓ ( x ℓ ) := ∑ j ∈ V ∖ { x ℓ } Q t ℓ ( j ∣ x ℓ ) v t ( x ) ∈ R L × K v_t(x) \in\mathbb{R}^{L\times K} v t ( x ) ∈ R L × K
v t ℓ j ( x ) = { Q ‾ t ℓ ( j ∣ x ) , j ≠ x ℓ , − λ t ℓ ( x ℓ ) , j = x ℓ , (37) v_t^{\ell j}(x) =
\begin{cases}
\overline{Q}_t^\ell(j|x), & j \ne x^\ell, \\
-\lambda_t^\ell(x^\ell), & j = x^\ell,
\end{cases} \tag*{(37)} v t ℓ j ( x ) = { Q t ℓ ( j ∣ x ) , − λ t ℓ ( x ℓ ) , j = x ℓ , j = x ℓ , (37) and the extended density μ t ( x , e ) = p t ( x ) exp ( − ∑ ℓ = 1 L ∑ j = 1 K e ℓ j ) \mu_t(x,e) = p_t(x)\exp\left(-\sum_{\ell=1}^{L}\sum_{j=1}^{K}e^{\ell j}\right) μ t ( x , e ) = p t ( x ) exp ( − ∑ ℓ = 1 L ∑ j = 1 K e ℓ j ) X × [ 0 , ∞ ) L × K \mathcal{X}\times[0,\infty)^{L\times K} X × [ 0 , ∞ ) L × K (36) as
∂ t μ t ( x , e ) = − ∇ e ⋅ ( v t ( x ) μ t ( x , e ) ) , (38) \partial_t\mu_t(x,e)=-\nabla_e\cdot\left(v_t(x)\mu_t(x,e)\right), \tag*{(38)} ∂ t μ t ( x , e ) = − ∇ e ⋅ ( v t ( x ) μ t ( x , e ) ) , (38) where ∇ e ⋅ \nabla_e\cdot ∇ e ⋅ e e e E t E_t E t (9) .
Setting
Notation and assumptions
Notation. By convention, we set ∏ i j = 1 \prod_i^j=1 ∏ i j = 1 i > j i>j i > j 0 ⋅ ( + ∞ ) 0\cdot(+\infty) 0 ⋅ ( + ∞ ) ℓ , h \ell,h ℓ , h j , k j,k j , k n , m n,m n , m
We recall that the we fix the state space X = V L \mathcal{X}=\mathcal{V}^L X = V L L ≥ 1 L\ge1 L ≥ 1 V = { 1 , … , K } \mathcal{V}=\{1,\ldots,K\} V = { 1 , … , K } X = V L \mathcal{X}=\mathcal{V}^L X = V L D = L K D=LK D = L K R D \mathbb{R}^D R D [ [ 1 , L ] ] × [ [ 1 , K ] ] [\![1,L]\!]\times[\![1,K]\!] [ [ 1 , L ] ] × [ [ 1 , K ] ] x ∈ X x\in\mathcal{X} x ∈ X ℓ ∈ { 1 , … , L } \ell\in\{1,\ldots,L\} ℓ ∈ { 1 , … , L } j ∈ V j\in\mathcal{V} j ∈ V x − ℓ x^{-\ell} x − ℓ ( x 1 , … , x ℓ − 1 , x ℓ + 1 , … , x L ) (x^1,\ldots,x^{\ell-1},x^{\ell+1},\ldots,x^L) ( x 1 , … , x ℓ − 1 , x ℓ + 1 , … , x L ) x ℓ → j x^{\ell\to j} x ℓ → j y y y y − ℓ = x − ℓ y^{-\ell}=x^{-\ell} y − ℓ = x − ℓ y ℓ = j y^\ell=j y ℓ = j
Setting and assumptions. We consider now an inhomogeneous rate matrix denoted by ( Q v ) v ∈ [ 0 , 1 ] (Q_v)_{v\in[0,1]} ( Q v ) v ∈ [ 0 , 1 ] i.e. , satisfying for any v ∈ [ 0 , 1 ] v\in[0,1] v ∈ [ 0 , 1 ] Q v ( y ∣ x ) ≥ 0 Q_v(y|x)\ge0 Q v ( y ∣ x ) ≥ 0 x ≠ y x\ne y x = y
Q v ( x ∣ x ) = − λ v ( x ) = − ∑ y ≠ x Q v ( y ∣ x ) . (39) Q_v(x|x)=-\lambda_v(x)=-\sum_{y\ne x}Q_v(y|x). \tag*{(39)} Q v ( x ∣ x ) = − λ v ( x ) = − y = x ∑ Q v ( y ∣ x ) . (39) We denote by p v p_v p v v v v
Assumption 1. (i) There exist L L L V \mathcal{V} V { ( Q v ℓ ) v ∈ [ 0 , 1 ] : ℓ ∈ { 1 , … , L } } \{(Q_v^\ell)_{v\in[0,1]}:\ell\in\{1,\ldots,L\}\} {( Q v ℓ ) v ∈ [ 0 , 1 ] : ℓ ∈ { 1 , … , L }} x , y ∈ X x,y\in\mathcal{X} x , y ∈ X
Q v ( y ∣ x ) = ∑ ℓ = 1 L Q v ℓ ( y ℓ ∣ x ℓ ) 1 { x − ℓ = y − ℓ } . (40) Q_v(y|x)=\sum_{\ell=1}^{L}Q_v^\ell(y^\ell|x^\ell)\mathbf{1}_{\{x^{-\ell}=y^{-\ell}\}}. \tag*{(40)} Q v ( y ∣ x ) = ℓ = 1 ∑ L Q v ℓ ( y ℓ ∣ x ℓ ) 1 { x − ℓ = y − ℓ } . (40) (ii) For every ℓ \ell ℓ x ℓ , x ~ ℓ ∈ V x^\ell,\widetilde{x}^\ell\in\mathcal{V} x ℓ , x ℓ ∈ V v ↦ Q v ℓ ( x ℓ ∣ x ~ ℓ ) v\mapsto Q_v^\ell(x^\ell|\widetilde{x}^\ell) v ↦ Q v ℓ ( x ℓ ∣ x ℓ ) [ 0 , 1 ] [0,1] [ 0 , 1 ]
(iii) There is no explosion, i.e. , ∑ x ∈ X p v ( x ) = 1 \sum_{x\in\mathcal{X}}p_v(x)=1 ∑ x ∈ X p v ( x ) = 1 v ∈ [ 0 , 1 ] v\in[0,1] v ∈ [ 0 , 1 ]
(iv) For any v ∈ [ 0 , 1 ] v\in[0,1] v ∈ [ 0 , 1 ] x ∈ X x\in\mathcal{X} x ∈ X p v ( x ) > 0 p_v(x)>0 p v ( x ) > 0
(v) For every x ∈ X x\in\mathcal{X} x ∈ X v ↦ p v ( x ) v\mapsto p_v(x) v ↦ p v ( x ) [ 0 , 1 ] [0,1] [ 0 , 1 ] v ∈ ( 0 , 1 ) v\in(0,1) v ∈ ( 0 , 1 )
∂ v p v ( x ) = ∑ ℓ = 1 L ∑ y − ℓ = x − ℓ y ℓ ≠ x ℓ p v ( y ) Q v ℓ ( x ℓ ∣ y ℓ ) − p v ( x ) λ v ( x ) . (41) \partial_vp_v(x)=\sum_{\ell=1}^{L}\sum_{\substack{y^{-\ell}=x^{-\ell}\\y^\ell\ne x^\ell}}p_v(y)Q_v^\ell(x^\ell|y^\ell)-p_v(x)\lambda_v(x). \tag*{(41)} ∂ v p v ( x ) = ℓ = 1 ∑ L y − ℓ = x − ℓ y ℓ = x ℓ ∑ p v ( y ) Q v ℓ ( x ℓ ∣ y ℓ ) − p v ( x ) λ v ( x ) . (41) The derivatives at 0 and 1 are one-sided.
Under Assumption 1, denote for any ℓ ∈ { 1 , … , L } \ell\in\{1,\ldots,L\} ℓ ∈ { 1 , … , L } x ℓ ∈ V x^\ell\in V x ℓ ∈ V
λ v ℓ ( x ℓ ) = ∑ j : j ≠ x ℓ Q ‾ v ℓ ( j ∣ x ℓ ) , (42) \lambda^\ell_v(x^\ell)=\sum_{j:j\ne x^\ell}\overline{Q}^\ell_v(j\mid x^\ell), \tag*{(42)} λ v ℓ ( x ℓ ) = j : j = x ℓ ∑ Q v ℓ ( j ∣ x ℓ ) , (42) and note that with this notation, λ v ( x ) = ∑ ℓ = 1 L λ v ℓ ( x ℓ ) \lambda_v(x)=\sum_{\ell=1}^{L}\lambda^\ell_v(x^\ell) λ v ( x ) = ∑ ℓ = 1 L λ v ℓ ( x ℓ ) Q v ( x ∣ x ) = − λ v ( x ) Q_v(x\mid x)=-\lambda_v(x) Q v ( x ∣ x ) = − λ v ( x )
For v ∈ [ 0 , 1 ] v\in[0,1] v ∈ [ 0 , 1 ] x , y ∈ X x,y\in\mathcal{X} x , y ∈ X
Q ‾ v ( y ∣ x ) = p v ( y ) p v ( x ) Q v ( x ∣ y ) . (43) \overline{Q}_v(y\mid x)=\frac{p_v(y)}{p_v(x)}Q_v(x\mid y). \tag*{(43)} Q v ( y ∣ x ) = p v ( x ) p v ( y ) Q v ( x ∣ y ) . (43) Under Assumption 1, the reverse rates are well-defined and satisfy, for distinct x , y x,y x , y
Q ‾ v ( y ∣ x ) = ∑ ℓ ∈ { 1 , … , L } Q ‾ v ℓ ( y ℓ ∣ x ℓ ) 1 { x − ℓ = y − ℓ } , \overline{Q}_v(y\mid x)=\sum_{\ell\in\{1,\ldots,L\}}\overline{Q}^\ell_v(y^\ell\mid x^\ell)\mathbf{1}_{\{x^{-\ell}=y^{-\ell}\}}, Q v ( y ∣ x ) = ℓ ∈ { 1 , … , L } ∑ Q v ℓ ( y ℓ ∣ x ℓ ) 1 { x − ℓ = y − ℓ } , Q ‾ v ℓ ( j ∣ x ) = p v ( y ) p v ( x ) Q v ℓ ( x ℓ ∣ j ) . (44) \overline{Q}^\ell_v(j\mid x)=\frac{p_v(y)}{p_v(x)}Q^\ell_v(x^\ell\mid j). \tag*{(44)} Q v ℓ ( j ∣ x ) = p v ( x ) p v ( y ) Q v ℓ ( x ℓ ∣ j ) . (44) In the second line, y − ℓ = x − ℓ y^{-\ell}=x^{-\ell} y − ℓ = x − ℓ y ℓ = j y^\ell=j y ℓ = j λ ‾ v ( x ) = ∑ ℓ = 1 L ∑ j : j ≠ x ℓ Q ‾ v ℓ ( j ∣ x ) \overline{\lambda}_v(x)=\sum_{\ell=1}^{L}\sum_{j:j\ne x^\ell}\overline{Q}^\ell_v(j\mid x) λ v ( x ) = ∑ ℓ = 1 L ∑ j : j = x ℓ Q v ℓ ( j ∣ x ) Q ‾ v ( x ∣ x ) = − λ ‾ v ( x ) \overline{Q}_v(x\mid x)=-\overline{\lambda}_v(x) Q v ( x ∣ x ) = − λ v ( x )
ERF sampler revisited
We complete the description of the residual dynamics, described in Appendix § 12 .§ 1 , mainly introducing new notations. For x ∈ X x\in\mathcal{X} x ∈ X v ∈ [ 0 , 1 ] v\in[0,1] v ∈ [ 0 , 1 ] X → R D \mathcal{X}\to\mathbb{R}^{D} X → R D x ∈ X x\in\mathcal{X} x ∈ X ℓ ∈ { 1 , … , L } \ell\in\{1,\ldots,L\} ℓ ∈ { 1 , … , L } j ∈ V j\in V j ∈ V
b v ℓ j ( x ) = − Q ‾ v ℓ ( j ∣ x ) 1 { j ≠ x ℓ } + λ v ℓ ( x ℓ ) 1 { j = x ℓ } . (45) b^{\ell j}_v(x)=-\overline{Q}^{\ell}_v(j\mid x)\mathbf{1}_{\{j\ne x^\ell\}}+\lambda^\ell_v(x^\ell)\mathbf{1}_{\{j=x^\ell\}}. \tag*{(45)} b v ℓ j ( x ) = − Q v ℓ ( j ∣ x ) 1 { j = x ℓ } + λ v ℓ ( x ℓ ) 1 { j = x ℓ } . (45) In addition for 0 ≤ u ≤ v ≤ 1 0\leq u\leq v\leq1 0 ≤ u ≤ v ≤ 1
Δ s ∣ t ( x ) = ∫ s t b u ( x ) d u . (46) \Delta_{s\mid t}(x)=\int_s^t b_u(x)\,\mathrm{d}u. \tag*{(46)} Δ s ∣ t ( x ) = ∫ s t b u ( x ) d u . (46) If the state remains x x x v v v u u u
E u = E v + Δ u ∣ v ( x ) . (47) E_u=E_v+\Delta_{u\mid v}(x). \tag*{(47)} E u = E v + Δ u ∣ v ( x ) . (47) For v > 0 v>0 v > 0 x ∈ X x\in\mathcal{X} x ∈ X η ∈ [ 0 , ∞ ) D \eta\in[0,\infty)^D η ∈ [ 0 , ∞ ) D j ≠ x ℓ j\ne x^\ell j = x ℓ
θ ℓ j ( v , x , η ) = sup { u ∈ [ 0 , v ] : ∫ u v Q ‾ τ ℓ ( j ∣ x ) d τ ≥ η ℓ j } . (48) \theta^{\ell j}(v,x,\eta)=\sup\left\{u\in[0,v]:\int_u^v\overline{Q}^{\ell}_{\tau}(j\mid x)\,\mathrm{d}\tau\geq\eta^{\ell j}\right\}. \tag*{(48)} θ ℓ j ( v , x , η ) = sup { u ∈ [ 0 , v ] : ∫ u v Q τ ℓ ( j ∣ x ) d τ ≥ η ℓ j } . (48) Set sup ∅ = − ∞ \sup\varnothing=-\infty sup ∅ = − ∞ θ ℓ j ( v , x , η ) = − ∞ \theta^{\ell j}(v,x,\eta)=-\infty θ ℓ j ( v , x , η ) = − ∞ j = x ℓ j=x^\ell j = x ℓ
Set T 0 ∗ = 1 T^*_0=1 T 0 ∗ = 1 X 1 ∼ p 1 X_1\sim p_1 X 1 ∼ p 1 E 1 ℓ j E^{\ell j}_1 E 1 ℓ j ∼ Exp ( 1 ) \sim\operatorname{Exp}(1) ∼ Exp ( 1 ) ( X T 0 ∗ , E T 0 ∗ ) = ( X 1 , E 1 ) (X_{T^*_0},E_{T^*_0})=(X_1,E_1) ( X T 0 ∗ , E T 0 ∗ ) = ( X 1 , E 1 )
Law ( X 1 , E 1 ) = p 1 ⊗ Exp ( 1 ) ⊗ D . (49) \operatorname{Law}(X_1,E_1)=p_1\otimes\operatorname{Exp}(1)^{\otimes D}. \tag*{(49)} Law ( X 1 , E 1 ) = p 1 ⊗ Exp ( 1 ) ⊗ D . (49) Suppose that the first n n n ( T n ∗ , X T n ∗ , E T n ∗ ) = ( v , x , η ) (T^*_n,X_{T^*_n},E_{T^*_n})=(v,x,\eta) ( T n ∗ , X T n ∗ , E T n ∗ ) = ( v , x , η ) v > 0 v>0 v > 0 Θ n + 1 = max ( ℓ , j ) θ ℓ j ( v , x , η ) \Theta_{n+1}=\max_{(\ell,j)}\theta^{\ell j}(v,x,\eta) Θ n + 1 = max ( ℓ , j ) θ ℓ j ( v , x , η ) Θ n + 1 = − ∞ \Theta_{n+1}=-\infty Θ n + 1 = − ∞ 0 0 0
0 < Θ n + 1 < v , argmax ( ℓ , j ) θ ℓ j ( v , x , η ) = { ( h , k ) } , Q ‾ Θ n + 1 h ( k ∣ x ) > 0 , (50) 0<\Theta_{n+1}<v,\qquad\underset{(\ell,j)}{\operatorname{argmax}}\ \theta^{\ell j}(v,x,\eta)=\{(h,k)\},\qquad\overline{Q}^{h}_{\Theta_{n+1}}(k\mid x)>0, \tag*{(50)} 0 < Θ n + 1 < v , ( ℓ , j ) argmax θ ℓ j ( v , x , η ) = {( h , k )} , Q Θ n + 1 h ( k ∣ x ) > 0 , (50) set
T n + 1 ∗ = Θ n + 1 , C n + 1 = ( h , k ) , Y n + 1 = x , T^*_{n+1}=\Theta_{n+1},\qquad C_{n+1}=(h,k),\qquad Y_{n+1}=x, T n + 1 ∗ = Θ n + 1 , C n + 1 = ( h , k ) , Y n + 1 = x , E T n + 1 ∗ = η + Δ T n + 1 ∗ ∣ v ( x ) , X T n + 1 ∗ = y , y h = k , y ℓ = x ℓ for ℓ ≠ h . (51) E_{T^*_{n+1}}=\eta+\Delta_{T^*_{n+1}\mid v}(x),\qquad X_{T^*_{n+1}}=y,\qquad y^h=k,\qquad y^\ell=x^\ell\ \text{for }\ell\ne h. \tag*{(51)} E T n + 1 ∗ = η + Δ T n + 1 ∗ ∣ v ( x ) , X T n + 1 ∗ = y , y h = k , y ℓ = x ℓ for ℓ = h . (51) Here Y n + 1 Y_{n+1} Y n + 1 C n + 1 C_{n+1} C n + 1
X u = X T n ⋆ , E u = E T n ⋆ + Δ u ∣ T n ⋆ ( X T n ⋆ ) , Θ n + 1 < u ≤ T n ⋆ . (52) X_u = X_{T_n^\star}, \qquad E_u = E_{T_n^\star} + \Delta_{u\mid T_n^\star}(X_{T_n^\star}), \qquad\Theta_{n+1} < u \leq T_n^\star. \tag*{(52)} X u = X T n ⋆ , E u = E T n ⋆ + Δ u ∣ T n ⋆ ( X T n ⋆ ) , Θ n + 1 < u ≤ T n ⋆ . (52) If a successful step does not exist, set its time and all subsequent step times to − ∞ -\infty − ∞ N ∞ = sup { n ≥ 0 : T n ⋆ ≥ 0 } N_\infty= \sup\{n \geq0 : T_n^\star\geq0\} N ∞ = sup { n ≥ 0 : T n ⋆ ≥ 0 }
Main result
Let O = ( 0 , ∞ ) D \mathcal{O} = (0,\infty)^D O = ( 0 , ∞ ) D r ∈ [ 0 , ∞ ) D r \in[0,\infty)^D r ∈ [ 0 , ∞ ) D u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ]
γ ( r ) = exp ( − ∑ ℓ , j r ℓ j ) , F u ( x , r ) = p u ( x ) γ ( r ) . (53) \gamma(r) = \exp\left(-\sum_{\ell,j} r^{\ell j}\right), \qquad F_u(x,r) = p_u(x)\gamma(r). \tag*{(53)} γ ( r ) = exp − ℓ , j ∑ r ℓ j , F u ( x , r ) = p u ( x ) γ ( r ) . (53) Lebesgue measure on O \mathcal{O} O d r dr d r
Theorem I.1. Assume Assumption 1 , define the reverse rates by (43) –(44) , and consider the EFR sampler described in Appendix § 18 .2.2. Then, almost surely, N ∞ < ∞ N_\infty< \infty N ∞ < ∞ [ 0 , 1 ] [0,1] [ 0 , 1 ] u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ]
Law ( X u , E u ) = p u ⊗ Exp ( 1 ) ⊗ D . (54) \operatorname{Law}(X_u,E_u) = p_u \otimes\operatorname{Exp}(1)^{\otimes D}. \tag*{(54)} Law ( X u , E u ) = p u ⊗ Exp ( 1 ) ⊗ D . (54) In addition, every u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ] P ( T n ⋆ = u for some n ) = 0 \mathbb{P}(T_n^\star= u\text{ for some }n) = 0 P ( T n ⋆ = u for some n ) = 0
For n ≥ 0 n \geq0 n ≥ 0 u ∈ [ 0 , 1 ) u \in[0,1) u ∈ [ 0 , 1 )
H u ( n ) = { T n ⋆ > u , T n + 1 ⋆ < u } . (55) H_u^{(n)} = \{T_n^\star> u,\ T_{n+1}^\star< u\}. \tag*{(55)} H u ( n ) = { T n ⋆ > u , T n + 1 ⋆ < u } . (55) For each Borel set B ⊆ [ 0 , ∞ ) D \mathcal{B} \subseteq[0,\infty)^D B ⊆ [ 0 , ∞ ) D
μ u ( n ) ( { x } × B ) = E [ 1 H u ( n ) 1 { X T n ⋆ = x } 1 { E T n ⋆ + Δ u ∣ T n ⋆ ( X T n ⋆ ) ∈ B } ] . (56) \mu_u^{(n)}(\{x\}\times\mathcal{B}) = \mathbb{E}\left[\mathbf{1}_{H_u^{(n)}}\mathbf{1}_{\{X_{T_n^\star}=x\}}\mathbf{1}_{\{E_{T_n^\star}+\Delta_{u\mid T_n^\star}(X_{T_n^\star})\in\mathcal{B}\}}\right]. \tag*{(56)} μ u ( n ) ({ x } × B ) = E [ 1 H u ( n ) 1 { X T n ⋆ = x } 1 { E T n ⋆ + Δ u ∣ T n ⋆ ( X T n ⋆ ) ∈ B } ] . (56) We prove the theorem by constructing densities f u ( n ) f_u^{(n)} f u ( n ) (49) is not conditioning on a fixed state. The conclusion concerns deterministic times; it imposes no independence at jump times and does not assert that X X X
Preliminaries
For x ∈ X x \in\mathcal{X} x ∈ X ℓ ∈ [ 1 , L ] \ell\in[1,L] ℓ ∈ [ 1 , L ] j ≠ x ℓ j \ne x^\ell j = x ℓ 0 ≤ u ≤ v ≤ 1 0 \leq u \leq v \leq1 0 ≤ u ≤ v ≤ 1
A u ∣ v ℓ ( x ) = ∫ u v λ τ ℓ ( x ℓ ) d τ , B u ∣ v ℓ j ( x ) = ∫ u v Q ‾ τ ℓ ( j ∣ x ) d τ . (57) A_{u\mid v}^{\ell}(x) = \int_u^v \lambda_\tau^\ell(x^\ell)\,d\tau,\qquad B_{u\mid v}^{\ell j}(x) = \int_u^v \overline{Q}_\tau^{\ell}(j\mid x)\,d\tau. \tag*{(57)} A u ∣ v ℓ ( x ) = ∫ u v λ τ ℓ ( x ℓ ) d τ , B u ∣ v ℓ j ( x ) = ∫ u v Q τ ℓ ( j ∣ x ) d τ . (57) Consequently,
Δ u ∣ v ℓ j ( x ) = { A u ∣ v ℓ ( x ) , j = x ℓ , − B u ∣ v ℓ j ( x ) , j ≠ x ℓ . (58) \Delta_{u\mid v}^{\ell j}(x) =
\begin{cases}
A_{u\mid v}^{\ell}(x), & j=x^\ell,\\
-B_{u\mid v}^{\ell j}(x), & j\ne x^\ell.
\end{cases} \tag*{(58)} Δ u ∣ v ℓ j ( x ) = { A u ∣ v ℓ ( x ) , − B u ∣ v ℓ j ( x ) , j = x ℓ , j = x ℓ . (58) Lemma 1. Under Assumption 1 , all forward and reverse off-diagonal rates and all exit rates are continuous and bounded on [ 0 , 1 ] [0,1] [ 0 , 1 ] v ∈ [ 0 , 1 ] v \in[0,1] v ∈ [ 0 , 1 ] x ∈ X x \in\mathcal{X} x ∈ X ℓ \ell ℓ y y y y − ℓ = x − ℓ y^{-\ell}=x^{-\ell} y − ℓ = x − ℓ y ℓ ≠ x ℓ y^\ell\ne x^\ell y ℓ = x ℓ
∂ v log p v ( x ) = λ ‾ v ( x ) − λ v ( x ) , (59) \partial_v \log p_v(x) = \overline{\lambda}_v(x)-\lambda_v(x), \tag*{(59)} ∂ v log p v ( x ) = λ v ( x ) − λ v ( x ) , (59) p v ( y ) Q ‾ v ℓ ( x ℓ ∣ y ) = p v ( x ) Q v ℓ ( y ℓ ∣ x ℓ ) , (60) p_v(y)\overline{Q}_v^\ell(x^\ell\mid y)=p_v(x)Q_v^\ell(y^\ell\mid x^\ell), \tag*{(60)} p v ( y ) Q v ℓ ( x ℓ ∣ y ) = p v ( x ) Q v ℓ ( y ℓ ∣ x ℓ ) , (60) ∑ y − ℓ = x − ℓ y ℓ ≠ x ℓ p v ( y ) Q ‾ v ℓ ( x ℓ ∣ y ) = p v ( x ) λ v ℓ ( x ℓ ) . (61) \sum_{\substack{y^{-\ell}=x^{-\ell}\\ y^\ell\ne x^\ell}}p_v(y)\overline{Q}_v^\ell(x^\ell\mid y)=p_v(x)\lambda_v^\ell(x^\ell). \tag*{(61)} y − ℓ = x − ℓ y ℓ = x ℓ ∑ p v ( y ) Q v ℓ ( x ℓ ∣ y ) = p v ( x ) λ v ℓ ( x ℓ ) . (61) In (59) , endpoint derivatives are one-sided. In particular,
Q ‾ v ℓ ( x ℓ ∣ y ) > 0 ⟹ λ v ℓ ( x ℓ ) > 0. (62) \overline{Q}_v^\ell(x^\ell\mid y)>0 \quad\Longrightarrow\quad\lambda_v^\ell(x^\ell)>0. \tag*{(62)} Q v ℓ ( x ℓ ∣ y ) > 0 ⟹ λ v ℓ ( x ℓ ) > 0. (62) For 0 ≤ u ≤ v ≤ 1 0 \le u \le v \le1 0 ≤ u ≤ v ≤ 1 x ∈ X x \in\mathcal{X} x ∈ X r , w ∈ [ 0 , ∞ ) D r,w \in[0,\infty)^D r , w ∈ [ 0 , ∞ ) D r = w + Δ u ∣ v ( x ) r = w + \Delta_{u|v}(x) r = w + Δ u ∣ v ( x )
p v ( x ) γ ( w ) = p u ( x ) γ ( r ) . (63) p_v(x)\gamma(w) = p_u(x)\gamma(r). \tag*{(63)} p v ( x ) γ ( w ) = p u ( x ) γ ( r ) . (63) Proof. Since X \mathcal{X} X p v ( x ) p_v(x) p v ( x ) [ 0 , 1 ] [0,1] [ 0 , 1 ] p ∗ = min v ∈ [ 0 , 1 ] , x ∈ X p v ( x ) p_* = \min_{v\in[0,1],x\in\mathcal{X}}p_v(x) p ∗ = min v ∈ [ 0 , 1 ] , x ∈ X p v ( x ) (44) , and of their finite sums follow. Applying (44) to the transition from y y y x x x
Q ‾ v ℓ ( x ℓ ∣ y ) = p v ( x ) p v ( y ) Q ‾ v ℓ ( y ℓ ∣ x ℓ ) . \overline{Q}^{\ell}_{v}(x^\ell|y) = \frac{p_v(x)}{p_v(y)}\overline{Q}^{\ell}_{v}(y^\ell|x^\ell). Q v ℓ ( x ℓ ∣ y ) = p v ( y ) p v ( x ) Q v ℓ ( y ℓ ∣ x ℓ ) . This proves (60) ; summing over the predecessor states gives (61) . Nonnegativity of the summands and positivity of p v ( x ) p_v(x) p v ( x ) (62) . Dividing (41) by p v ( x ) p_v(x) p v ( x ) (44) proves (59) .
For 0 ≤ u ≤ v ≤ 1 0 \le u \le v \le1 0 ≤ u ≤ v ≤ 1 (45) and (59) give
∑ ℓ , j Δ u ∣ v ℓ j ( x ) = ∫ u v ( λ τ ( x ) − λ ‾ τ ( x ) ) d τ = − ∫ u v ∂ τ log p τ ( x ) d τ = log p u ( x ) p v ( x ) . (64) \begin{aligned}
\sum_{\ell,j}\Delta^{\ell j}_{u|v}(x) = \int_u^v\left(\lambda_\tau(x)-\overline{\lambda}_\tau(x)\right)\,\mathrm{d}\tau\\
&= -\int_u^v \partial_\tau\log p_\tau(x)\,\mathrm{d}\tau= \log\frac{p_u(x)}{p_v(x)}. \tag*{(64)}
\end{aligned} ℓ , j ∑ Δ u ∣ v ℓ j ( x ) = ∫ u v ( λ τ ( x ) − λ τ ( x ) ) d τ = − ∫ u v ∂ τ log p τ ( x ) d τ = log p v ( x ) p u ( x ) . (64) Since w = r − Δ u ∣ v ( x ) w = r-\Delta_{u|v}(x) w = r − Δ u ∣ v ( x ) (53) yields
p v ( x ) γ ( w ) = p v ( x ) γ ( r ) exp ( ∑ ℓ , j Δ u ∣ v ℓ j ( x ) ) = p u ( x ) γ ( r ) . p_v(x)\gamma(w) = p_v(x)\gamma(r)\exp\left(\sum_{\ell,j}\Delta^{\ell j}_{u|v}(x)\right) = p_u(x)\gamma(r). p v ( x ) γ ( w ) = p v ( x ) γ ( r ) exp ℓ , j ∑ Δ u ∣ v ℓ j ( x ) = p u ( x ) γ ( r ) . Lemma 2. Let α < β \alpha< \beta α < β ω : [ α , β ] → [ 0 , ∞ ) \omega:[\alpha,\beta]\to[0,\infty) ω : [ α , β ] → [ 0 , ∞ ) Ψ ( v ) = ∫ α v ω ( τ ) d τ \Psi(v)=\int_\alpha^v\omega(\tau)\,\mathrm{d}\tau Ψ ( v ) = ∫ α v ω ( τ ) d τ ρ ( ζ ) = inf { v ∈ [ α , β ] : Ψ ( v ) ≥ ζ } \rho(\zeta)=\inf\{v\in[\alpha,\beta]:\Psi(v)\ge\zeta\} ρ ( ζ ) = inf { v ∈ [ α , β ] : Ψ ( v ) ≥ ζ } 0 < ζ < Ψ ( β ) 0<\zeta<\Psi(\beta) 0 < ζ < Ψ ( β )
(i) The set Ψ ( { v ∈ [ α , β ] : ω ( v ) = 0 } ) \Psi(\{v\in[\alpha,\beta]:\omega(v)=0\}) Ψ ({ v ∈ [ α , β ] : ω ( v ) = 0 }) ζ ∈ ( 0 , Ψ ( β ) ) \zeta\in(0,\Psi(\beta)) ζ ∈ ( 0 , Ψ ( β )) Ψ ( v ) = ζ \Psi(v)=\zeta Ψ ( v ) = ζ ( α , β ) (\alpha,\beta) ( α , β ) ω ( v ) > 0 \omega(v)>0 ω ( v ) > 0 v 0 ∈ ( α , β ) v_0\in(\alpha,\beta) v 0 ∈ ( α , β ) ω ( v 0 ) > 0 \omega(v_0)>0 ω ( v 0 ) > 0 Ψ − 1 ( { Ψ ( v 0 ) } ) = { v 0 } \Psi^{-1}(\{\Psi(v_0)\})=\{v_0\} Ψ − 1 ({ Ψ ( v 0 )}) = { v 0 }
(ii) For every nonnegative Borel χ \chi χ [ 0 , Ψ ( β ) ] [0,\Psi(\beta)] [ 0 , Ψ ( β )]
∫ α β χ ( Ψ ( v ) ) ω ( v ) d v = ∫ 0 Ψ ( β ) χ ( ζ ) d ζ . (65) \int_\alpha^\beta\chi(\Psi(v))\omega(v)\,\mathrm{d}v = \int_0^{\Psi(\beta)}\chi(\zeta)\,\mathrm{d}\zeta. \tag*{(65)} ∫ α β χ ( Ψ ( v )) ω ( v ) d v = ∫ 0 Ψ ( β ) χ ( ζ ) d ζ . (65) For every nonnegative Borel φ \varphi φ [ α , β ] [\alpha,\beta] [ α , β ]
∫ α β φ ( v ) ω ( v ) d v = ∫ 0 Ψ ( β ) φ ( ρ ( ζ ) ) d ζ . (66) \int_\alpha^\beta\varphi(v)\omega(v)\,\mathrm{d}v = \int_0^{\Psi(\beta)}\varphi(\rho(\zeta))\,\mathrm{d}\zeta. \tag*{(66)} ∫ α β φ ( v ) ω ( v ) d v = ∫ 0 Ψ ( β ) φ ( ρ ( ζ )) d ζ . (66) (iii) If υ 1 , υ 2 : [ α , β ] → R \upsilon_1,\upsilon_2:[\alpha,\beta]\to\mathbb{R} υ 1 , υ 2 : [ α , β ] → R { ( υ 1 ( v ) , υ 2 ( v ) ) : v ∈ [ α , β ] } \{(\upsilon_1(v),\upsilon_2(v)):v\in[\alpha,\beta]\} {( υ 1 ( v ) , υ 2 ( v )) : v ∈ [ α , β ]} ( 0 , Ψ ( β ) ) (0,\Psi(\beta)) ( 0 , Ψ ( β )) Ψ ( β ) = 0 \Psi(\beta)=0 Ψ ( β ) = 0
Proof. Fix ϵ > 0 \epsilon>0 ϵ > 0 O ϵ = { v ∈ [ α , β ] : ω ( v ) < ϵ } O_\epsilon=\{v\in[\alpha,\beta]:\omega(v)<\epsilon\} O ϵ = { v ∈ [ α , β ] : ω ( v ) < ϵ } J J J Ψ \Psi Ψ length ( Ψ ( J ) ) ≤ ϵ length ( J ) \operatorname{length}(\Psi(J))\le\epsilon\operatorname{length}(J) length ( Ψ ( J )) ≤ ϵ length ( J ) ω \omega ω ϵ ( β − α ) \epsilon(\beta-\alpha) ϵ ( β − α ) ϵ \epsilon ϵ 0 , Ψ ( β ) 0,\Psi(\beta) 0 , Ψ ( β ) v ∈ ( α , β ) v\in(\alpha,\beta) v ∈ ( α , β ) Ψ ( v ) = ζ \Psi(v)=\zeta Ψ ( v ) = ζ ω ( v ) > 0 \omega(v)>0 ω ( v ) > 0 v 1 < v 2 v_1<v_2 v 1 < v 2 Ψ \Psi Ψ [ v 1 , v 2 ] [v_1,v_2] [ v 1 , v 2 ] ω ( v 1 ) = ω ( v 2 ) = 0 \omega(v_1)=\omega(v_2)=0 ω ( v 1 ) = ω ( v 2 ) = 0 ω ( v 0 ) > 0 \omega(v_0)>0 ω ( v 0 ) > 0
For continuous χ \chi χ Φ ( ζ ) = ∫ 0 ζ χ ( ζ ′ ) d ζ ′ \Phi(\zeta)=\int_0^\zeta\chi(\zeta')\,\mathrm{d}\zeta' Φ ( ζ ) = ∫ 0 ζ χ ( ζ ′ ) d ζ ′ ( Φ ∘ Ψ ) ′ ( v ) = χ ( Ψ ( v ) ) ω ( v ) (\Phi\circ\Psi)'(v)=\chi(\Psi(v))\omega(v) ( Φ ∘ Ψ ) ′ ( v ) = χ ( Ψ ( v )) ω ( v ) (65) . Equality against continuous functions identifies the two finite measures, so the identity extends to nonnegative Borel χ \chi χ α < v < β \alpha< v<\beta α < v < β ω ( v ) > 0 \omega(v)>0 ω ( v ) > 0 ρ ( Ψ ( v ) ) = v \rho(\Psi(v))=v ρ ( Ψ ( v )) = v (65) to χ ( ζ ) = φ ( ρ ( ζ ) ) \chi(\zeta)=\varphi(\rho(\zeta)) χ ( ζ ) = φ ( ρ ( ζ )) ζ \zeta ζ (66) .
For (iii), let Λ \Lambda Λ [ α , β ] [\alpha,\beta] [ α , β ] m m m ( β − α ) / m (\beta-\alpha)/m ( β − α ) / m 2 Λ ( β − α ) / m 2\Lambda(\beta-\alpha)/m 2Λ ( β − α ) / m m m m 4 Λ 2 ( β − α ) 2 / m 4\Lambda^2(\beta-\alpha)^2/m 4 Λ 2 ( β − α ) 2 / m m → ∞ m\to\infty m → ∞
For a numerical pair c ∈ [ 1 , L ] × [ 1 , K ] c\in[1,L]\times[1,K] c ∈ [ 1 , L ] × [ 1 , K ]
B c = { r ∈ [ 0 , ∞ ) D : r c = 0 , r d > 0 for every d ≠ c } . (67) B_c=\{r\in[0,\infty)^D:r^c=0,\ r^d>0\text{ for every }d\ne c\}. \tag*{(67)} B c = { r ∈ [ 0 , ∞ ) D : r c = 0 , r d > 0 for every d = c } . (67) Let ι c : ( 0 , ∞ ) D − 1 → B c \iota_c:(0,\infty)^{D-1}\to B_c ι c : ( 0 , ∞ ) D − 1 → B c c c c σ c \sigma_c σ c B c B_c B c
∫ B c φ ( w ) σ c ( d w ) = ∫ ( 0 , ∞ ) D − 1 φ ( ι c ( ζ ) ) d ζ (68) \int_{B_c}\varphi(w)\,\sigma_c(dw)=\int_{(0,\infty)^{D-1}}\varphi(\iota_c(\zeta))\,d\zeta \tag*{(68)} ∫ B c φ ( w ) σ c ( d w ) = ∫ ( 0 , ∞ ) D − 1 φ ( ι c ( ζ )) d ζ (68) for nonnegative Borel φ \varphi φ σ c ( d w ) = δ 0 ( d w c ) ∏ d ≠ c d w d \sigma_c(dw)=\delta_0(dw^c)\prod_{d\ne c}dw^d σ c ( d w ) = δ 0 ( d w c ) ∏ d = c d w d B c B_c B c x ∈ X x\in X x ∈ X
c ℓ ( x ) = the unique pair ( ℓ , j ) with j = x ℓ , O x = O ∪ ⋃ ℓ , j : j ≠ x ℓ B ( ℓ , j ) . (69) c_\ell(x)=\text{the unique pair }(\ell,j)\text{ with }j=x^\ell,\qquad
O_x=O\cup\bigcup_{\ell,j:j\ne x^\ell}B_{(\ell,j)}. \tag*{(69)} c ℓ ( x ) = the unique pair ( ℓ , j ) with j = x ℓ , O x = O ∪ ℓ , j : j = x ℓ ⋃ B ( ℓ , j ) . (69) Thus r c ℓ ( x ) = ⟨ x ℓ , r ℓ ⟩ r^{c_\ell(x)}=\langle x^\ell,r^\ell\rangle r c ℓ ( x ) = ⟨ x ℓ , r ℓ ⟩ u ∈ [ 0 , 1 ) u\in[0,1) u ∈ [ 0 , 1 ) x ∈ X x\in X x ∈ X r ∈ O x r\in O_x r ∈ O x
N u ( x ) = { r ∈ O x : ⟨ x h , r h ⟩ > A u ∣ 1 h ( x ) for every h } , (70) N_u(x)=\{r\in O_x:\langle x^h,r^h\rangle>A^h_{u|1}(x)\text{ for every }h\}, \tag*{(70)} N u ( x ) = { r ∈ O x : ⟨ x h , r h ⟩ > A u ∣1 h ( x ) for every h } , (70) ρ ℓ ( u , x , r ) = inf { v ∈ [ u , 1 ] : A u ∣ v ℓ ( x ) ≥ ⟨ x ℓ , r ℓ ⟩ } , inf ∅ = + ∞ . (71) \rho_\ell(u,x,r)=\inf\{v\in[u,1]:A^\ell_{u|v}(x)\ge\langle x^\ell,r^\ell\rangle\},\qquad\inf\varnothing=+\infty. \tag*{(71)} ρ ℓ ( u , x , r ) = inf { v ∈ [ u , 1 ] : A u ∣ v ℓ ( x ) ≥ ⟨ x ℓ , r ℓ ⟩} , inf ∅ = + ∞. (71) Define also
R u ℓ ( x ) = { r ∈ O x : ∃ v ∈ ( u , 1 ) such that λ v ℓ ( x ℓ ) > 0 , \mathcal{R}^\ell_u(x)=\left\{r\in O_x:\exists v\in(u,1)\text{ such that }
\lambda^\ell_v(x^\ell)>0,\right. R u ℓ ( x ) = { r ∈ O x : ∃ v ∈ ( u , 1 ) such that λ v ℓ ( x ℓ ) > 0 , ⟨ x ℓ , r ℓ ⟩ = A u ∣ v ℓ ( x ) , ⟨ x h , r h ⟩ > A u ∣ v h ( x ) ( h ≠ ℓ ) , r d − Δ u ∣ v d ( x ) > 0 ( d ≠ c ℓ ( x ) ) } (72) \left.
\begin{aligned}
\langle x^\ell,r^\ell\rangle&=A^\ell_{u|v}(x),\\
\langle x^h,r^h\rangle&>A^h_{u|v}(x) &&(h\ne\ell),\\
r^d-\Delta^d_{u|v}(x)&>0 &&(d\ne c_\ell(x))
\end{aligned}
\right\} \tag*{(72)} ⟨ x ℓ , r ℓ ⟩ ⟨ x h , r h ⟩ r d − Δ u ∣ v d ( x ) = A u ∣ v ℓ ( x ) , > A u ∣ v h ( x ) > 0 ( h = ℓ ) , ( d = c ℓ ( x )) ⎭ ⎬ ⎫ (72) and
R u ( x ) = ⋃ ℓ = 1 L R u ℓ ( x ) . (73) \mathcal{R}_u(x)=\bigcup_{\ell=1}^{L}\mathcal{R}^\ell_u(x). \tag*{(73)} R u ( x ) = ℓ = 1 ⋃ L R u ℓ ( x ) . (73) For r ∈ R u ℓ ( x ) r\in\mathcal{R}^\ell_u(x) r ∈ R u ℓ ( x )
v = ρ ℓ ( u , x , r ) , c = c ℓ ( x ) , w = r − Δ u ∣ v ( x ) ∈ B c . (74) v=\rho_\ell(u,x,r),\qquad c=c_\ell(x),\qquad w=r-\Delta_{u|v}(x)\in B_c. \tag*{(74)} v = ρ ℓ ( u , x , r ) , c = c ℓ ( x ) , w = r − Δ u ∣ v ( x ) ∈ B c . (74) Lemma 3. Assume Assumption 1.
(i) The maps in (71) are Borel in ( u , r ) (u,r) ( u , r ) x x x (70) and (72) , as sets of ( u , r ) (u,r) ( u , r ) R u ℓ ( x ) \mathcal{R}^\ell_u(x) R u ℓ ( x ) N u ( x ) N_u(x) N u ( x )
(ii) For each fixed u ∈ [ 0 , 1 ) u\in[0,1) u ∈ [ 0 , 1 ) x ∈ X x\in X x ∈ X
Leb D ( O ∖ ( N u ( x ) ∪ R u ( x ) ) ) = 0. (75) \operatorname{Leb}_D\bigl(O\setminus(N_u(x)\cup\mathcal{R}_u(x))\bigr)=0. \tag*{(75)} Leb D ( O ∖ ( N u ( x ) ∪ R u ( x )) ) = 0. (75) For each such u , x u,x u , x d = ( h , j ) d=(h,j) d = ( h , j ) j ≠ x h j\ne x^h j = x h Q ‾ u h ( j ∣ x ) > 0 \overline{Q}^h_u(j|x)>0 Q u h ( j ∣ x ) > 0
σ d ( B d ∖ ( N u ( x ) ∪ R u ( x ) ) ) = 0. (76) \sigma_d\bigl(B_d\setminus(N_u(x)\cup\mathcal{R}_u(x))\bigr)=0. \tag*{(76)} σ d ( B d ∖ ( N u ( x ) ∪ R u ( x )) ) = 0. (76) Equivalently, the exceptional set is null for Lebesgue measure in the D − 1 D-1 D − 1 d d d x x x d = ( h , j ) d=(h,j) d = ( h , j )
∫ 0 1 ∫ B d 1 { r ∉ N u ( x ) ∪ R u ( x ) } Q ‾ u h ( j ∣ x ) σ d ( d r ) d u = 0. (77) \int_0^1\int_{B_d}\mathbf{1}_{\{r\notin N_u(x)\cup\mathcal{R}_u(x)\}}\overline{Q}^h_u(j|x)\,\sigma_d(dr)\,du=0. \tag*{(77)} ∫ 0 1 ∫ B d 1 { r ∈ / N u ( x ) ∪ R u ( x )} Q u h ( j ∣ x ) σ d ( d r ) d u = 0. (77) (iii) Fix 0 ≤ u < v < 1 0\le u<v<1 0 ≤ u < v < 1 x ∈ X x\in X x ∈ X ℓ \ell ℓ w ∈ B c ℓ ( x ) w\in B_{c_\ell(x)} w ∈ B c ℓ ( x ) λ v ℓ ( x ℓ ) > 0 \lambda^\ell_v(x^\ell)>0 λ v ℓ ( x ℓ ) > 0 r = w + Δ u ∣ v ( x ) r=w+\Delta_{u|v}(x) r = w + Δ u ∣ v ( x ) r ∈ O r\in O r ∈ O [ u , v ) [u,v) [ u , v ) r ∈ R u ℓ ( x ) r\in\mathcal{R}^\ell_u(x) r ∈ R u ℓ ( x ) ρ ℓ ( u , x , r ) = v \rho_\ell(u,x,r)=v ρ ℓ ( u , x , r ) = v
same reconstruction conclusion holds if r ∈ B d r \in\mathcal{B}_d r ∈ B d d = ( h , j ) d=(h,j) d = ( h , j ) Q ˉ u h ( j ∣ x ) > 0 \bar{Q}^{h}_{u}(j\mid x)>0 Q ˉ u h ( j ∣ x ) > 0 v v v u u u d d d
(iv) Conversely, if r ∈ R u ℓ ( x ) r \in\mathcal{R}^{\ell}_{u}(x) r ∈ R u ℓ ( x ) (74) gives the only pair ( v , ℓ ) (v,\ell) ( v , ℓ ) x x x c ℓ ( x ) c_{\ell}(x) c ℓ ( x ) ( u , x , r ) (u,x,r) ( u , x , r ) r ∈ B d r \in\mathcal{B}_d r ∈ B d Q ˉ u h ( j ∣ x ) > 0 \bar{Q}^{h}_{u}(j\mid x)>0 Q ˉ u h ( j ∣ x ) > 0 d = ( h , j ) d=(h,j) d = ( h , j )
Proof. For τ ∈ [ 0 , 1 ] \tau\in[0,1] τ ∈ [ 0 , 1 ] r ∈ O x r \in\mathcal{O}_x r ∈ O x
{ ( u , r ) : ρ ℓ ( u , x , r ) ≤ τ } = { ( u , r ) : u ≤ τ , ⟨ x ℓ , r ℓ ⟩ ≤ A u ∣ τ ℓ ( x ) } . \{(u,r):\rho_{\ell}(u,x,r)\leq\tau\}=\{(u,r):u\leq\tau,\ \langle x^{\ell},r^{\ell}\rangle\leq A^{\ell}_{u\mid\tau}(x)\}. {( u , r ) : ρ ℓ ( u , x , r ) ≤ τ } = {( u , r ) : u ≤ τ , ⟨ x ℓ , r ℓ ⟩ ≤ A u ∣ τ ℓ ( x )} . The right-hand side is Borel because the integrated rate is continuous. If a witness v v v (72) , positivity of λ v ℓ ( x ℓ ) \lambda^{\ell}_{v}(x^{\ell}) λ v ℓ ( x ℓ ) ρ ℓ ( u , x , r ) \rho_{\ell}(u,x,r) ρ ℓ ( u , x , r ) ρ h ( u , x , r ) > v \rho_h(u,x,r)>v ρ h ( u , x , r ) > v h ≠ ℓ h\neq\ell h = ℓ { u < ρ ℓ ( u , x , r ) < 1 } \{u<\rho_{\ell}(u,x,r)<1\} { u < ρ ℓ ( u , x , r ) < 1 } N u ( x ) \mathcal{N}_u(x) N u ( x )
Fix u , x u,x u , x ℓ \ell ℓ
{ 0 , A u ∣ 1 ℓ ( x ) } ∪ { A u ∣ v ℓ ( x ) : v ∈ [ u , 1 ] , λ v ℓ ( x ℓ ) = 0 } \{0,A^{\ell}_{u\mid1}(x)\}\cup\{A^{\ell}_{u\mid v}(x):v\in[u,1],\ \lambda^{\ell}_{v}(x^{\ell})=0\} { 0 , A u ∣ 1 ℓ ( x )} ∪ { A u ∣ v ℓ ( x ) : v ∈ [ u , 1 ] , λ v ℓ ( x ℓ ) = 0 } for the coordinate ⟨ x ℓ , r ℓ ⟩ \langle x^{\ell},r^{\ell}\rangle ⟨ x ℓ , r ℓ ⟩ ℓ ≠ h \ell\neq h ℓ = h
( ⟨ x ℓ , r ℓ ⟩ , ⟨ x h , r h ⟩ ) ∈ { ( A u ∣ v ℓ ( x ) , A u ∣ v h ( x ) ) : v ∈ [ u , 1 ] } . (\langle x^{\ell},r^{\ell}\rangle,\langle x^{h},r^{h}\rangle)\in\{(A^{\ell}_{u\mid v}(x),A^{h}_{u\mid v}(x)):v\in[u,1]\}. (⟨ x ℓ , r ℓ ⟩ , ⟨ x h , r h ⟩) ∈ {( A u ∣ v ℓ ( x ) , A u ∣ v h ( x )) : v ∈ [ u , 1 ]} . Their planar measure is zero by Lemma 2(iii). Fubini and the finite number of indices show that the excluded set is null in O \mathcal{O} O N u ( x ) \mathcal{N}_u(x) N u ( x ) v v v ℓ \ell ℓ ( u , 1 ) (u,1) ( u , 1 ) λ v ℓ ( x ℓ ) > 0 \lambda^{\ell}_{v}(x^{\ell})>0 λ v ℓ ( x ℓ ) > 0 h ≠ ℓ h\neq\ell h = ℓ ⟨ x h , r h ⟩ − A u ∣ v h ( x ) > 0 \langle x^h,r^h\rangle-A^h_{u\mid v}(x)>0 ⟨ x h , r h ⟩ − A u ∣ v h ( x ) > 0 d = ( h , j ) d=(h,j) d = ( h , j )
r d − Δ u ∣ v d ( x ) = r d + B u ∣ v h j ( x ) > 0. r^d-\Delta^d_{u\mid v}(x)=r^d+B^{hj}_{u\mid v}(x)>0. r d − Δ u ∣ v d ( x ) = r d + B u ∣ v hj ( x ) > 0. The remaining equality at c ℓ ( x ) c_{\ell}(x) c ℓ ( x ) R u ℓ ( x ) \mathcal{R}^{\ell}_{u}(x) R u ℓ ( x ) (75) .
On B d \mathcal{B}_d B d d d d x x x L L L σ d \sigma_d σ d r d = 0 r^d=0 r d = 0 d = ( h , j ) d=(h,j) d = ( h , j ) Q ˉ u h ( j ∣ x ) > 0 \bar{Q}^{h}_{u}(j\mid x)>0 Q ˉ u h ( j ∣ x ) > 0 B u ∣ v h j ( x ) > 0 B^{hj}_{u\mid v}(x)>0 B u ∣ v hj ( x ) > 0 v > u v>u v > u w d > 0 w^d>0 w d > 0 (76) . Joint Borel measurability and Tonelli, restricted first to bounded coordinate boxes if necessary, give (77) .
For (iii), ⟨ x ℓ , r ℓ ⟩ = A u ∣ v ℓ ( x ) \langle x^{\ell},r^{\ell}\rangle=A^{\ell}_{u\mid v}(x) ⟨ x ℓ , r ℓ ⟩ = A u ∣ v ℓ ( x ) λ v ℓ ( x ℓ ) > 0 \lambda^{\ell}_{v}(x^{\ell})>0 λ v ℓ ( x ℓ ) > 0 v v v ρ ℓ ( u , x , r ) = v \rho_{\ell}(u,x,r)=v ρ ℓ ( u , x , r ) = v h ≠ ℓ h\neq\ell h = ℓ ⟨ x h , w h ⟩ \langle x^h,w^h\rangle ⟨ x h , w h ⟩ ⟨ x h , r h ⟩ > A u ∣ v h ( x ) \langle x^h,r^h\rangle>A^h_{u\mid v}(x) ⟨ x h , r h ⟩ > A u ∣ v h ( x ) ρ h ( u , x , r ) > v \rho_h(u,x,r)>v ρ h ( u , x , r ) > v c ℓ ( x ) c_{\ell}(x) c ℓ ( x ) v v v v v v r d = 0 r^d=0 r d = 0 Q ˉ u h ( j ∣ x ) > 0 \bar{Q}^{h}_{u}(j\mid x)>0 Q ˉ u h ( j ∣ x ) > 0 d d d B u ∣ τ h j ( x ) > 0 B^{hj}_{u\mid\tau}(x)>0 B u ∣ τ hj ( x ) > 0 τ ∈ ( u , v ) \tau\in(u,v) τ ∈ ( u , v ) u u u
For (iv), the definitions give the stated w w w ρ h \rho_h ρ h (74) .
Lemma 4
Lemma 4. Under Assumption 1, every finite construction in (48) –(52) , including its success and failure events, is Borel measurable in its initial condition. After a hit satisfying (50) at which only its triggering residual is zero, every residual is strictly positive for a sufficiently short interval in decreasing time after the update, and no immediate further hit occurs. Proof. For fixed x x x ( ℓ , j ) (\ell,j) ( ℓ , j ) u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ]
{ ( v , η ) : θ ℓ j ( v , x , η ) ≥ u } = { ( v , η ) : v ≥ u , ∫ u v Q ‾ τ ℓ ( j ∣ x ) d τ ≥ η ℓ j } \{(v,\eta):\theta^{\ell j}(v,x,\eta)\ge u\}=\left\{(v,\eta):v\ge u,\ \int_u^v\overline{Q}^{\ell}_\tau(j|x)\,\mathrm{d}\tau\ge\eta^{\ell j}\right\} {( v , η ) : θ ℓ j ( v , x , η ) ≥ u } = { ( v , η ) : v ≥ u , ∫ u v Q τ ℓ ( j ∣ x ) d τ ≥ η ℓ j } on η ℓ j > 0 \eta^{\ell j}>0 η ℓ j > 0 (55) –(56) .
Suppose the state changes from y y y x x x v v v y − ℓ = x − ℓ y^{-\ell}=x^{-\ell} y − ℓ = x − ℓ y ℓ ≠ x ℓ y^\ell\ne x^\ell y ℓ = x ℓ Q ‾ v ℓ ( x ℓ ∣ y ) > 0 \overline{Q}^{\ell}_v(x^\ell|y)>0 Q v ℓ ( x ℓ ∣ y ) > 0 (62) gives λ v ℓ ( x ℓ ) > 0 \lambda^\ell_v(x^\ell)>0 λ v ℓ ( x ℓ ) > 0 λ τ ℓ ( x ℓ ) > 0 \lambda^\ell_\tau(x^\ell)>0 λ τ ℓ ( x ℓ ) > 0 v v v
Change of variables
Lemma 5. Assume Assumption 1 and (49) . Define, for every u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] x ∈ X x\in\mathsf{X} x ∈ X r ∈ O x r\in\mathcal{O}_x r ∈ O x
f u ( 0 ) ( x , r ) : = F u ( x , r ) 1 N u ( x ) ( r ) . (78) f_u^{(0)}(x,r):=F_u(x,r)\mathbf{1}_{\mathcal{N}_u(x)}(r). \tag*{(78)} f u ( 0 ) ( x , r ) := F u ( x , r ) 1 N u ( x ) ( r ) . (78) Then, for every u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] x ∈ X x\in\mathsf{X} x ∈ X B ⊆ O B\subseteq\mathcal{O} B ⊆ O
μ u ( 0 ) ( { x } × B ) = ∫ B f u ( 0 ) ( x , r ) d r . (79) \mu_u^{(0)}(\{x\}\times B)=\int_B f_u^{(0)}(x,r)\,\mathrm{d}r. \tag*{(79)} μ u ( 0 ) ({ x } × B ) = ∫ B f u ( 0 ) ( x , r ) d r . (79) The measure μ u ( 0 ) \mu_u^{(0)} μ u ( 0 ) X × ( [ 0 , ∞ ) D ∖ O ) \mathsf{X}\times([0,\infty)^D\setminus\mathcal{O}) X × ([ 0 , ∞ ) D ∖ O )
For every x ∈ X x\in\mathsf{X} x ∈ X d = ( h , j ) ∈ [ 1 , L ] × [ 1 , K ] d=(h,j)\in[1,L]\times[1,K] d = ( h , j ) ∈ [ 1 , L ] × [ 1 , K ] j ≠ x h j\ne x^h j = x h ψ : ( 0 , 1 ) × B d → [ 0 , ∞ ] \psi:(0,1)\times\mathcal{B}_d\to[0,\infty] ψ : ( 0 , 1 ) × B d → [ 0 , ∞ ]
E [ 1 { T 1 ∈ ( 0 , 1 ) , Y 1 = x , C 1 = d } ψ ( T 1 , E T 1 ) ] = ∫ 0 1 ∫ B d ψ ( v , w ) Q ‾ v h ( j ∣ x ) f v ( 0 ) ( x , w ) σ d ( d w ) d v . (80) \mathbb{E}\left[\mathbf{1}_{\{T_1\in(0,1),Y_1=x,C_1=d\}}\psi(T_1,E_{T_1})\right]
=\int_0^1\int_{\mathcal{B}_d}\psi(v,w)\overline{Q}^{h}_v(j|x)f_v^{(0)}(x,w)\,\sigma_d(\mathrm{d}w)\,\mathrm{d}v. \tag*{(80)} E [ 1 { T 1 ∈ ( 0 , 1 ) , Y 1 = x , C 1 = d } ψ ( T 1 , E T 1 ) ] = ∫ 0 1 ∫ B d ψ ( v , w ) Q v h ( j ∣ x ) f v ( 0 ) ( x , w ) σ d ( d w ) d v . (80) The first hit, when present on [ 0 , 1 ] [0,1] [ 0 , 1 ] (50) almost surely and has only its firing residual equal to zero. For each fixed s ∈ [ 0 , 1 ] s\in[0,1] s ∈ [ 0 , 1 ] s s s
Proof. Fix u , x u,x u , x H u ( 0 ) ∩ { X u = x } H_u^{(0)}\cap\{X_u=x\} H u ( 0 ) ∩ { X u = x } r r r
η = r − Δ u ∣ 1 ( x ) . \eta=r-\Delta_{u|1}(x). η = r − Δ u ∣1 ( x ) . For a passive index c ℓ ( x ) c\ell(x) c ℓ ( x ) η c ℓ ( x ) \eta^{c\ell(x)} η c ℓ ( x ) ⟨ x ℓ , r ℓ ⟩ > A u ∣ 1 ℓ ( x ) \langle x^\ell,r^\ell\rangle>A^\ell_{u|1}(x) ⟨ x ℓ , r ℓ ⟩ > A u ∣1 ℓ ( x ) ( ℓ , j ) (\ell,j) ( ℓ , j )
η ℓ j = r ℓ j + B u ∣ 1 ℓ j ( x ) . \eta^{\ell j}=r^{\ell j}+B^{\ell j}_{u|1}(x). η ℓ j = r ℓ j + B u ∣1 ℓ j ( x ) . If r r r τ ∈ [ u , 1 ] \tau\in[u,1] τ ∈ [ u , 1 ] r ℓ j + B u ∣ τ ℓ j ( x ) r^{\ell j}+B^{\ell j}_{u|\tau}(x) r ℓ j + B u ∣ τ ℓ j ( x ) I D I_D I D
p 1 ( x ) γ ( r − Δ u ∣ 1 ( x ) ) 1 N u ( x ) ( r ) = F u ( x , r ) 1 N u ( x ) ( r ) , p_1(x)\gamma(r-\Delta_{u|1}(x))\mathbf{1}_{\mathcal{N}_u(x)}(r)=F_u(x,r)\mathbf{1}_{\mathcal{N}_u(x)}(r), p 1 ( x ) γ ( r − Δ u ∣1 ( x )) 1 N u ( x ) ( r ) = F u ( x , r ) 1 N u ( x ) ( r ) , by (63) . Initial residuals are strictly positive, and no hit down to u u u (79) and the assertion about boundary mass.
We first exclude the exceptional first hits. Condition on X 1 = x X_1=x X 1 = x d = ( h , j ) d=(h,j) d = ( h , j ) v v v
E 1 d = B v ∣ 1 h j ( x ) . E_1^d=B^{hj}_{v|1}(x). E 1 d = B v ∣1 hj ( x ) . As a function of v v v C 1 C^1 C 1 − Q ‾ v h ( j ∣ x ) -\overline{Q}^{h}_v(j|x) − Q v h ( j ∣ x ) s ∈ [ 0 , 1 ] s \in[0,1] s ∈ [ 0 , 1 ] E 1 d = B s ∣ 1 h j ( x ) E^d_1 = B^{hj}_{s|1}(x) E 1 d = B s ∣1 hj ( x ) d = ( h , j ) d=(h,j) d = ( h , j ) a = ( ℓ , k ) a=(\ell,k) a = ( ℓ , k )
( E 1 d , E 1 a ) ∈ { ( B v ∣ 1 h j ( x ) , B v ∣ 1 ℓ k ( x ) ) : v ∈ [ 0 , 1 ] } . \left(E^d_1,E^a_1\right)\in\left\{\left(B^{hj}_{v|1}(x),B^{\ell k}_{v|1}(x)\right):v\in[0,1]\right\}. ( E 1 d , E 1 a ) ∈ { ( B v ∣1 hj ( x ) , B v ∣1 ℓ k ( x ) ) : v ∈ [ 0 , 1 ] } . This set is planar-null by Lemma 2(iii), whereas the pair of independent exponentials has a Lebesgue density. A finite union over indices and states excludes all these cases. The passive residuals increase from positive values, so all residuals other than the unique firing residual are positive at this hit.
Now fix x x x d = ( h , j ) d=(h,j) d = ( h , j ) w = ι d ( ζ ) w=\iota_d(\zeta) w = ι d ( ζ )
U x , d : = { ( v , ζ ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) D − 1 : Q ‾ v h ( j ∣ x ) > 0 , ⟨ x ℓ , w ℓ ⟩ > A v ∣ 1 ℓ ( x ) for every ℓ ∈ [ 1 , L ] } . (81) U_{x,d}:=\left\{(v,\zeta)\in(0,1)\times(0,\infty)^{D-1}:\overline{Q}^{h}_{v}(j|x)>0,\ \langle x^\ell,w^\ell\rangle>A^\ell_{v|1}(x)\text{ for every }\ell\in[1,L]\right\}. \tag*{(81)} U x , d := { ( v , ζ ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) D − 1 : Q v h ( j ∣ x ) > 0 , ⟨ x ℓ , w ℓ ⟩ > A v ∣1 ℓ ( x ) for every ℓ ∈ [ 1 , L ] } . (81) Define U : U x , d → ( 0 , ∞ ) D U:U_{x,d}\to(0,\infty)^D U : U x , d → ( 0 , ∞ ) D
U ( v , ζ ) = w − Δ v ∣ 1 ( x ) . (82) U(v,\zeta)=w-\Delta_{v|1}(x). \tag*{(82)} U ( v , ζ ) = w − Δ v ∣1 ( x ) . (82) In particular, its d d d B v ∣ 1 h j ( x ) B^{hj}_{v|1}(x) B v ∣1 hj ( x ) w a + B v ∣ 1 a ( x ) w^a+B^a_{v|1}(x) w a + B v ∣1 a ( x ) B v ∣ 1 a = B v ∣ 1 ℓ k B^a_{v|1}=B^{\ell k}_{v|1} B v ∣1 a = B v ∣1 ℓ k a = ( ℓ , k ) a=(\ell,k) a = ( ℓ , k ) ⟨ x ℓ , w ℓ ⟩ − A v ∣ 1 ℓ ( x ) \langle x^\ell,w^\ell\rangle-A^\ell_{v|1}(x) ⟨ x ℓ , w ℓ ⟩ − A v ∣1 ℓ ( x ) (81) .
Order the target coordinates as ( d , − d ) (d,-d) ( d , − d ) ( v , ζ ) (v,\zeta) ( v , ζ ) ∂ v Δ v ∣ 1 ( x ) = − b v ( x ) \partial_v\Delta_{v|1}(x)=-b_v(x) ∂ v Δ v ∣1 ( x ) = − b v ( x )
D U ( v , ζ ) = ( − Q ‾ v h ( j ∣ x ) 0 1 × ( D − 1 ) b v − d ( x ) I D − 1 ) , ∣ det D U ( v , ζ ) ∣ = Q ‾ v h ( j ∣ x ) . (83) DU(v,\zeta)=\begin{pmatrix}
-\overline{Q}^{h}_{v}(j|x) & 0_{1\times(D-1)}\\
b^{-d}_{v}(x) & I_{D-1}
\end{pmatrix},
\qquad\lvert\det DU(v,\zeta)\rvert=\overline{Q}^{h}_{v}(j|x). \tag*{(83)} D U ( v , ζ ) = ( − Q v h ( j ∣ x ) b v − d ( x ) 0 1 × ( D − 1 ) I D − 1 ) , ∣ det D U ( v , ζ )∣ = Q v h ( j ∣ x ) . (83) Here b v − d b^{-d}_{v} b v − d d d d
The map U U U d d d v v v Q ‾ v h ( j ∣ x ) > 0 \overline{Q}^{h}_{v}(j|x)>0 Q v h ( j ∣ x ) > 0 v v v w = η + Δ v ∣ 1 ( x ) w=\eta+\Delta_{v|1}(x) w = η + Δ v ∣1 ( x ) ζ \zeta ζ v v v w a w^a w a v v v (50) yields one point of (81) . Therefore the change-of-variables theorem applied to U U U (80) as
∫ U x , d ψ ( v , ι d ζ ) p 1 ( x ) γ ( U ( v , ζ ) ) Q ‾ v h ( j ∣ x ) d v d ζ . \int_{U_{x,d}}\psi(v,\iota_d\zeta)p_1(x)\gamma(U(v,\zeta))\overline{Q}^{h}_{v}(j|x)\,dv\,d\zeta. ∫ U x , d ψ ( v , ι d ζ ) p 1 ( x ) γ ( U ( v , ζ )) Q v h ( j ∣ x ) d v d ζ . The map is C 1 C^1 C 1 (63) ,
p 1 ( x ) γ ( w − Δ v ∣ 1 ( x ) ) ∏ ℓ = 1 L 1 { ⟨ x ℓ , w ℓ ⟩ > A v ∣ 1 ℓ ( x ) } = f v ( 0 ) ( x , w ) . p_1(x)\gamma\left(w-\Delta_{v|1}(x)\right)\prod_{\ell=1}^{L}\mathbf{1}_{\{\langle x^\ell,w^\ell\rangle>A^\ell_{v|1}(x)\}}=f^{(0)}_v(x,w). p 1 ( x ) γ ( w − Δ v ∣1 ( x ) ) ℓ = 1 ∏ L 1 {⟨ x ℓ , w ℓ ⟩ > A v ∣1 ℓ ( x )} = f v ( 0 ) ( x , w ) . Inserting this identity, using (68) , and observing that the integrand is zero when Q ‾ v h ( j ∣ x ) = 0 \overline{Q}^{h}_{v}(j|x)=0 Q v h ( j ∣ x ) = 0 (80) . The exceptional hits have already been excluded.
Lemma 6. Assume Assumption 1 and fix x ∈ X x\in\mathsf{X} x ∈ X ℓ ∈ [ 1 , L ] \ell\in[1,L] ℓ ∈ [ 1 , L ] ( v , w ) ↦ g v ℓ ( x , w ) (v,w)\mapsto g^\ell_v(x,w) ( v , w ) ↦ g v ℓ ( x , w ) ( 0 , 1 ) × B c ℓ ( x ) (0,1)\times\mathcal{B}_{c_\ell}(x) ( 0 , 1 ) × B c ℓ ( x )
∑ ℓ = 1 L ∫ 0 1 ∫ B c ℓ ( x ) g v ℓ ( x , w ) σ c ℓ ( x ) ( d w ) d v < ∞ , g v ℓ ( x , w ) = 0 if λ v ℓ ( x ℓ ) = 0. (84) \sum_{\ell=1}^{L}\int_0^1\int_{\mathcal{B}_{c_\ell}(x)}g^\ell_v(x,w)\,\sigma_{c_\ell(x)}(dw)\,dv<\infty,\qquad g^\ell_v(x,w)=0\text{ if }\lambda^\ell_v(x^\ell)=0. \tag*{(84)} ℓ = 1 ∑ L ∫ 0 1 ∫ B c ℓ ( x ) g v ℓ ( x , w ) σ c ℓ ( x ) ( d w ) d v < ∞ , g v ℓ ( x , w ) = 0 if λ v ℓ ( x ℓ ) = 0. (84) For data ( v , ℓ , w ) (v,\ell,w) ( v , ℓ , w ) v v v x x x w w w (47) toward 0 0 0 θ ( v , x , w ) \theta(v,x,w) θ ( v , x , w ) C ( v , x , w ) C(v,x,w) C ( v , x , w ) θ = − ∞ \theta=-\infty θ = − ∞ C C C
For every u ∈ [ 0 , 1 ) u \in[0,1) u ∈ [ 0 , 1 ) r ∈ O x r \in\mathcal{O}_x r ∈ O x
h u ( x , r ) : = { g v ℓ ( x , w ) λ v ℓ ( x ℓ ) , r ∈ R u ℓ ( x ) , v = ρ ℓ ( u , x , r ) , w = r − Δ u ∣ v ( x ) , 0 , r ∉ R u ( x ) . (85) h_u(x,r) :=
\begin{cases}
\dfrac{g_v^\ell(x,w)}{\lambda_v^\ell(x^\ell)}, & r \in\mathcal{R}_u^\ell(x),\quad v=\rho_\ell(u,x,r),\quad w=r-\Delta_{u|v}(x),\\
0, & r \notin\mathcal{R}_u(x).
\end{cases} \tag*{(85)} h u ( x , r ) := ⎩ ⎨ ⎧ λ v ℓ ( x ℓ ) g v ℓ ( x , w ) , 0 , r ∈ R u ℓ ( x ) , v = ρ ℓ ( u , x , r ) , w = r − Δ u ∣ v ( x ) , r ∈ / R u ( x ) . (85) This defines its values on B d B_d B d
(i) For every deterministic u ∈ [ 0 , 1 ) u \in[0,1) u ∈ [ 0 , 1 ) B ⊆ O B \subseteq\mathcal{O} B ⊆ O
∫ B h u ( x , r ) d r = ∑ ℓ = 1 L ∫ u 1 ∫ B c ℓ ( x ) 1 { w + Δ u ∣ v ( x ) ∈ B } g v ℓ ( x , w ) σ c ℓ ( x ) ( d w ) d v . (86) \int_B h_u(x,r)\,\mathrm{d}r
=
\sum_{\ell=1}^{L}\int_u^1\int_{B_{c_\ell(x)}}\mathbf{1}_{\{w+\Delta_{u|v}(x)\in B\}}g_v^\ell(x,w)\,\sigma_{c_\ell(x)}(\mathrm{d}w)\,\mathrm{d}v. \tag*{(86)} ∫ B h u ( x , r ) d r = ℓ = 1 ∑ L ∫ u 1 ∫ B c ℓ ( x ) 1 { w + Δ u ∣ v ( x ) ∈ B } g v ℓ ( x , w ) σ c ℓ ( x ) ( d w ) d v . (86) Only source times v v v u < v < 1 u<v<1 u < v < 1 u u u B B B
(ii) For every active index d = ( h , j ) d=(h,j) d = ( h , j ) j ≠ x h j\ne x^h j = x h ψ : ( 0 , 1 ) × B d → [ 0 , ∞ ] \psi:(0,1)\times B_d\to[0,\infty] ψ : ( 0 , 1 ) × B d → [ 0 , ∞ ]
∑ ℓ = 1 L ∫ 0 1 ∫ B c ℓ ( x ) g v ℓ ( x , w ) 1 { 0 < θ ( v , x , w ) < v , C ( v , x , w ) = d } × ψ ( θ ( v , x , w ) , w + Δ θ ( v , x , w ) ∣ v ( x ) ) σ c ℓ ( x ) ( d w ) d v = ∫ 0 1 ∫ B d ψ ( u , r ) Q ‾ u h ( j ∣ x ) h u ( x , r ) σ d ( d r ) d u . (87) \begin{aligned}
&\sum_{\ell=1}^{L}\int_0^1\int_{B_{c_\ell(x)}}g_v^\ell(x,w)\mathbf{1}_{\{0<\theta(v,x,w)<v,\ C(v,x,w)=d\}}\\
&\qquad{}\times\psi\bigl(\theta(v,x,w),w+\Delta_{\theta(v,x,w)|v}(x)\bigr)\,\sigma_{c_\ell(x)}(\mathrm{d}w)\,\mathrm{d}v\\
&=\int_0^1\int_{B_d}\psi(u,r)\overline{Q}^{h}_u(j|x)h_u(x,r)\,\sigma_d(\mathrm{d}r)\,\mathrm{d}u. \tag*{(87)}
\end{aligned} ℓ = 1 ∑ L ∫ 0 1 ∫ B c ℓ ( x ) g v ℓ ( x , w ) 1 { 0 < θ ( v , x , w ) < v , C ( v , x , w ) = d } × ψ ( θ ( v , x , w ) , w + Δ θ ( v , x , w ) ∣ v ( x ) ) σ c ℓ ( x ) ( d w ) d v = ∫ 0 1 ∫ B d ψ ( u , r ) Q u h ( j ∣ x ) h u ( x , r ) σ d ( d r ) d u . (87) Thus u u u 0 < u < v < 1 0<u<v<1 0 < u < v < 1
(iii) With respect to the finite source measures in (84) , a next hit, when present, is almost surely unique and has positive incoming speed. For every fixed s ∈ [ 0 , 1 ] s\in[0,1] s ∈ [ 0 , 1 ] s s s (86) –(87) .
Proof. Fix an index ℓ \ell ℓ c = c ℓ ( x ) c=c_\ell(x) c = c ℓ ( x ) (84) , it is enough to use v ∈ ( 0 , 1 ) v\in(0,1) v ∈ ( 0 , 1 ) λ v ℓ ( x ℓ ) > 0 \lambda_v^\ell(x^\ell)>0 λ v ℓ ( x ℓ ) > 0
For a fixed such v v v D − 1 D-1 D − 1 w − c w^{-c} w − c c c c v v v d = ( h , j ) d=(h,j) d = ( h , j ) u u u
w d = B u ∣ v h j ( x ) , 0 ≤ u < v . (88) w^d=B_{u|v}^{hj}(x),\qquad0\le u<v. \tag*{(88)} w d = B u ∣ v hj ( x ) , 0 ≤ u < v . (88) For fixed v v v w d w^d w d Q ‾ u h ( j ∣ x ) = 0 \overline{Q}^{h}_u(j|x)=0 Q u h ( j ∣ x ) = 0 s < v s<v s < v w d = B s ∣ v h j ( x ) w^d=B_{s|v}^{hj}(x) w d = B s ∣ v hj ( x ) s = v s=v s = v w − c w^{-c} w − c v v v s = 0 s=0 s = 0
We prove (i). Fix u ∈ [ 0 , 1 ) u\in[0,1) u ∈ [ 0 , 1 ) w = ι c ( ζ ) w=\iota_c(\zeta) w = ι c ( ζ ) ζ ∈ ( 0 , ∞ ) D − 1 \zeta\in(0,\infty)^{D-1} ζ ∈ ( 0 , ∞ ) D − 1
{ ( v , ζ ) : u < v < 1 , λ v ℓ ( x ℓ ) > 0 , ζ > 0 , ι c ( ζ ) + Δ u ∣ v ( x ) ∈ O } , (89) \{(v,\zeta):u<v<1,\ \lambda_v^\ell(x^\ell)>0,\ \zeta>0,\ \iota_c(\zeta)+\Delta_{u|v}(x)\in\mathcal{O}\}, \tag*{(89)} {( v , ζ ) : u < v < 1 , λ v ℓ ( x ℓ ) > 0 , ζ > 0 , ι c ( ζ ) + Δ u ∣ v ( x ) ∈ O } , (89) consider
V u ( v , ζ ) = ι c ( ζ ) + Δ u ∣ v ( x ) . V_u(v,\zeta)=\iota_c(\zeta)+\Delta_{u|v}(x). V u ( v , ζ ) = ι c ( ζ ) + Δ u ∣ v ( x ) . Order the target coordinates as ( c , − c ) (c,-c) ( c , − c )
D V u ( v , ζ ) = ( λ v ℓ ( x ℓ ) 0 1 × ( D − 1 ) b v − c ( x ) I D − 1 ) , ∣ det D V u ∣ = λ v ℓ ( x ℓ ) . (90) DV_u(v,\zeta)=\begin{pmatrix}\lambda_v^\ell(x^\ell)&0_{1\times(D-1)}\\ b_v^{-c}(x)&I_{D-1}\end{pmatrix},\qquad\lvert\det DV_u\rvert=\lambda_v^\ell(x^\ell). \tag*{(90)} D V u ( v , ζ ) = ( λ v ℓ ( x ℓ ) b v − c ( x ) 0 1 × ( D − 1 ) I D − 1 ) , ∣ det D V u ∣ = λ v ℓ ( x ℓ ) . (90) In particular, the component r c = A u ∣ v ℓ ( x ) r^c=A_{u\mid v}^\ell(x) r c = A u ∣ v ℓ ( x ) λ v ℓ ( x ℓ ) \lambda_v^\ell(x^\ell) λ v ℓ ( x ℓ ) v v v b v a ( x ) b_v^a(x) b v a ( x ) v v v
Lemma 3(iii)–(iv) shows that V u V_u V u (89) , with image R u ℓ ( x ) ∩ O \mathcal{R}_u^\ell(x)\cap O R u ℓ ( x ) ∩ O (74) . Images for distinct ℓ \ell ℓ u u u v v v (90) , and then summing over ℓ \ell ℓ (86) . The source measure is zero where λ v ℓ ( x ℓ ) = 0 \lambda_v^\ell(x^\ell)=0 λ v ℓ ( x ℓ ) = 0
We next prove (ii). Fix an active d = ( h , j ) d=(h,j) d = ( h , j ) c c c
0 < u < v < 1 , ζ ∈ ( 0 , ∞ ) D − 2 , 0<u<v<1,\qquad\zeta\in(0,\infty)^{D-2}, 0 < u < v < 1 , ζ ∈ ( 0 , ∞ ) D − 2 , indexed in the last factor by ( [ 1 , L ] × [ 1 , K ] ) ∖ { c , d } ([1,L]\times[1,K])\setminus\{c,d\} ([ 1 , L ] × [ 1 , K ]) ∖ { c , d } r r r w w w
r d = 0 , r c = A u ∣ v ℓ ( x ) , r a = ζ a ( a ∉ { c , d } ) , r^d=0,\qquad r^c=A_{u\mid v}^\ell(x),\qquad r^a=\zeta^a\quad(a\notin\{c,d\}), r d = 0 , r c = A u ∣ v ℓ ( x ) , r a = ζ a ( a ∈ / { c , d }) , w c = 0 , w d = B u ∣ v h j ( x ) , w a = ζ a − Δ u ∣ v a ( x ) ( a ∉ { c , d } ) . (91) w^c=0,\qquad w^d=B_{u\mid v}^{hj}(x),\qquad w^a=\zeta^a-\Delta_{u\mid v}^a(x)\quad(a\notin\{c,d\}). \tag*{(91)} w c = 0 , w d = B u ∣ v hj ( x ) , w a = ζ a − Δ u ∣ v a ( x ) ( a ∈ / { c , d }) . (91) Restrict these variables to the open set on which
λ v ℓ ( x ℓ ) > 0 , Q ‾ u h ( j ∣ x ) > 0 , w d > 0 ( a ≠ c ) . (92) \lambda_v^\ell(x^\ell)>0,\qquad\overline{Q}_u^h(j\mid x)>0,\qquad w^d>0\quad(a\ne c). \tag*{(92)} λ v ℓ ( x ℓ ) > 0 , Q u h ( j ∣ x ) > 0 , w d > 0 ( a = c ) . (92) On this set, r ∈ B d r\in B_d r ∈ B d (88) determines u u u r a r^a r a
There are two changes of variables on (92) . First define
U 1 ( v , u , ζ ) = ( v , w − c ) , U_1(v,u,\zeta)=(v,w^{-c}), U 1 ( v , u , ζ ) = ( v , w − c ) , ordering the target as ( v , w d , w − { c , d } ) (v,w^d,w^{-\{c,d\}}) ( v , w d , w − { c , d } )
∂ v B u ∣ v h j ( x ) = Q ‾ v h ( j ∣ x ) , ∂ u B u ∣ v h j ( x ) = − Q ‾ u h ( j ∣ x ) , \partial_v B_{u\mid v}^{hj}(x)=\overline{Q}_v^h(j\mid x),\qquad\partial_u B_{u\mid v}^{hj}(x)=-\overline{Q}_u^h(j\mid x), ∂ v B u ∣ v hj ( x ) = Q v h ( j ∣ x ) , ∂ u B u ∣ v hj ( x ) = − Q u h ( j ∣ x ) , and ∂ v Δ u ∣ v a = b v a \partial_v\Delta_{u\mid v}^a=b_v^a ∂ v Δ u ∣ v a = b v a ∂ u Δ u ∣ v a = − b u a \partial_u\Delta_{u\mid v}^a=-b_u^a ∂ u Δ u ∣ v a = − b u a
D U 1 = ( 1 0 0 Q ‾ v h ( j ∣ x ) − Q ‾ u h ( j ∣ x ) 0 − b v − { c , d } ( x ) b u − { c , d } ( x ) I D − 2 ) . (93) DU_1=\begin{pmatrix}
1&0&0\\
\overline{Q}_v^h(j\mid x)&-\overline{Q}_u^h(j\mid x)&0\\
-b_v^{-\{c,d\}}(x)&b_u^{-\{c,d\}}(x)&I_{D-2}
\end{pmatrix}. \tag*{(93)} D U 1 = 1 Q v h ( j ∣ x ) − b v − { c , d } ( x ) 0 − Q u h ( j ∣ x ) b u − { c , d } ( x ) 0 0 I D − 2 . (93) Expansion in the first row and then in the first row of the remaining minor gives
∣ det D U 1 ∣ = Q ‾ u h ( j ∣ x ) . (94) \lvert\det DU_1\rvert=\overline{Q}_u^h(j\mid x). \tag*{(94)} ∣ det D U 1 ∣ = Q u h ( j ∣ x ) . (94) For a fixed image ( v , w − c ) (v,w^{-c}) ( v , w − c ) w d = B u ∣ v h j ( x ) w^d=B_{u\mid v}^{hj}(x) w d = B u ∣ v hj ( x ) u u u u u u ζ \zeta ζ U 1 U_1 U 1
Second define
U 2 ( v , u , ζ ) = ( u , r − d ) , U_2(v,u,\zeta)=(u,r^{-d}), U 2 ( v , u , ζ ) = ( u , r − d ) , ordering the target as ( u , r c , r − { c , d } ) (u,r^c,r^{-\{c,d\}}) ( u , r c , r − { c , d } )
D U 2 = ( 0 1 0 λ v ℓ ( x ℓ ) − λ u ℓ ( x ℓ ) 0 0 0 I D − 2 ) , ∣ det D U 2 ∣ = λ v ℓ ( x ℓ ) . (95) DU_2=\begin{pmatrix}
0&1&0\\
\lambda_v^\ell(x^\ell)&-\lambda_u^\ell(x^\ell)&0\\
0&0&I_{D-2}
\end{pmatrix},\qquad\lvert\det DU_2\rvert=\lambda_v^\ell(x^\ell). \tag*{(95)} D U 2 = 0 λ v ℓ ( x ℓ ) 0 1 − λ u ℓ ( x ℓ ) 0 0 0 I D − 2 , ∣ det D U 2 ∣ = λ v ℓ ( x ℓ ) . (95) Indeed, the determinant of the upper-left 2 × 2 2 \times2 2 × 2 − λ v ℓ ( x ℓ ) -\lambda_v^\ell(x^\ell) − λ v ℓ ( x ℓ ) U 2 U_2 U 2 v v v ( u , x , r ) (u,x,r) ( u , x , r ) w w w D = 2 D=2 D = 2 I 0 I_0 I 0 ζ \zeta ζ
Applying these two substitutions in succession transforms
g v ℓ ( x , w ) d v σ c ( d w ) on hits through d g_v^\ell(x,w)\,\mathrm{d}v\,\sigma_c(\mathrm{d}w)\quad\text{on hits through }d g v ℓ ( x , w ) d v σ c ( d w ) on hits through d into
Q ‾ u h ( j ∣ x ) λ v ℓ ( x ℓ ) g v ℓ ( x , w ) d u σ d ( d r ) , r ∈ R u ℓ ( x ) ∩ B d , Q ‾ u h ( j ∣ x ) > 0. (96) \frac{\overline{Q}_u^h(j\mid x)}{\lambda_v^\ell(x^\ell)}g_v^\ell(x,w)\,\mathrm{d}u\,\sigma_d(\mathrm{d}r),\qquad r\in\mathcal{R}_u^\ell(x)\cap B_d,\quad\overline{Q}_u^h(j\mid x)>0. \tag*{(96)} λ v ℓ ( x ℓ ) Q u h ( j ∣ x ) g v ℓ ( x , w ) d u σ d ( d r ) , r ∈ R u ℓ ( x ) ∩ B d , Q u h ( j ∣ x ) > 0. (96) Both maps are C 1 C^1 C 1 U 2 U_2 U 2 (96) with its time coordinate. Images for distinct ℓ \ell ℓ (96) over ℓ \ell ℓ (87) with the pointwise definition (85) . Where Q ‾ u h ( j ∣ x ) = 0 \overline{Q}_u^h(j\mid x)=0 Q u h ( j ∣ x ) = 0 v v v
Induction on the number of jumps
Definition 1. Assume Assumption 1. Let f u ( x , r ) f_u(x,r) f u ( x , r ) { ( u , x , r ) : 0 ≤ u < 1 , x ∈ X , r ∈ O x } \{(u,x,r): 0\le u<1,\ x\in\mathcal{X},\ r\in\mathcal{O}_x\} {( u , x , r ) : 0 ≤ u < 1 , x ∈ X , r ∈ O x }
For each such ( u , x , r ) (u,x,r) ( u , x , r )
( J f ) u ( x , r ) : = { 1 λ v ℓ ( x ℓ ) ∑ y ∈ X ℓ y ℓ ≠ x ℓ Q ‾ v ℓ ( x ℓ ∣ y ℓ ) f v ( y , w ) , r ∈ R u ℓ ( x ) , 0 , r ∉ R u ( x ) . (97) (\mathcal{J}f)_u(x,r):=
\begin{cases}
\frac{1}{\lambda_v^\ell(x^\ell)}
\sum_{\substack{y\in\mathcal{X}^{\ell}\\y^\ell\ne x^\ell}}
\overline{Q}_v^\ell(x^\ell\mid y^\ell)f_v(y,w), & r\in\mathcal{R}_u^\ell(x),\\
0, & r\notin\mathcal{R}_u(x).
\end{cases} \tag*{(97)} ( J f ) u ( x , r ) := ⎩ ⎨ ⎧ λ v ℓ ( x ℓ ) 1 ∑ y ∈ X ℓ y ℓ = x ℓ Q v ℓ ( x ℓ ∣ y ℓ ) f v ( y , w ) , 0 , r ∈ R u ℓ ( x ) , r ∈ / R u ( x ) . (97) In the first line, ℓ \ell ℓ r ∈ R u ℓ ( x ) r\in\mathcal{R}_u^\ell(x) r ∈ R u ℓ ( x ) v = ρ ℓ ( u , x , r ) v=\rho_\ell(u,x,r) v = ρ ℓ ( u , x , r ) w = r − Δ u ∣ v ( x ) w=r-\Delta_u|v(x) w = r − Δ u ∣ v ( x ) y y y c = c ℓ ( x ) c=c_\ell(x) c = c ℓ ( x ) y y y w ∈ B c ⊂ O y w\in B_c\subset\mathcal{O}_y w ∈ B c ⊂ O y (97) is on the specified domain. All denominators in its first line are strictly positive by (72) .
Proposition 1. Assume Assumption 1 and (49) . Define f ( 0 ) f^{(0)} f ( 0 ) (78) , with its values on all of O x \mathcal{O}_x O x
f ( n + 1 ) : = J f ( n ) , n ∈ N ∪ { 0 } . (98) f^{(n+1)}:=\mathcal{J}f^{(n)},\qquad n\in\mathbb{N}\cup\{0\}. \tag*{(98)} f ( n + 1 ) := J f ( n ) , n ∈ N ∪ { 0 } . (98) Then every f ( n ) f^{(n)} f ( n ) n ≥ 0 n\ge0 n ≥ 0 u ∈ [ 0 , 1 ) u\in[0,1) u ∈ [ 0 , 1 ) x ∈ X x\in\mathcal{X} x ∈ X B ⊆ O B\subseteq\mathcal{O} B ⊆ O
μ u ( n ) ( { x } × B ) = ∫ B f u ( n ) ( x , r ) d r , μ u ( n ) ( X × ( [ 0 , ∞ ) D ∖ O ) ) = 0. (99) \mu_u^{(n)}(\{x\}\times B)=\int_B f_u^{(n)}(x,r)\,\mathrm{d}r,\qquad
\mu_u^{(n)}\bigl(\mathcal{X}\times([0,\infty)^D\setminus\mathcal{O})\bigr)=0. \tag*{(99)} μ u ( n ) ({ x } × B ) = ∫ B f u ( n ) ( x , r ) d r , μ u ( n ) ( X × ([ 0 , ∞ ) D ∖ O ) ) = 0. (99) Moreover, for every active index d = ( h , j ) d=(h,j) d = ( h , j ) x x x ψ : ( 0 , 1 ) × B d → [ 0 , ∞ ] \psi:(0,1)\times B_d\to[0,\infty] ψ : ( 0 , 1 ) × B d → [ 0 , ∞ ]
E [ 1 { T n + 1 ∈ ( 0 , 1 ) , Y n + 1 = x , C n + 1 = d } ψ ( T n + 1 , E T n + 1 ∗ ) ] = ∫ 0 1 ∫ B d ψ ( v , w ) Q ‾ v h ( j ∣ x ) f v ( n ) ( x , w ) σ d ( d w ) d v . (100) \mathbb{E}\left[\mathbf{1}_{\{T_{n+1}\in(0,1),\,Y_{n+1}=x,\,C_{n+1}=d\}}\psi(T_{n+1},E_{T_{n+1}}^*)\right]
=
\int_0^1\int_{B_d}\psi(v,w)\overline{Q}_v^h(j\mid x)f_v^{(n)}(x,w)\,\sigma_d(\mathrm{d}w)\,\mathrm{d}v. \tag*{(100)} E [ 1 { T n + 1 ∈ ( 0 , 1 ) , Y n + 1 = x , C n + 1 = d } ψ ( T n + 1 , E T n + 1 ∗ ) ] = ∫ 0 1 ∫ B d ψ ( v , w ) Q v h ( j ∣ x ) f v ( n ) ( x , w ) σ d ( d w ) d v . (100) For each fixed finite index, a hit reached by the recursion is almost surely either absent or satisfies (50) ; only its firing residual is zero. For every fixed s ∈ [ 0 , 1 ] s\in[0,1] s ∈ [ 0 , 1 ] n ≥ 1 n\ge1 n ≥ 1 n n n s s s
Proof. The function f ( 0 ) f^{(0)} f ( 0 ) (97) , and positivity of the denominator on its domain show inductively that every f ( n ) f^{(n)} f ( n )
For n = 0 n=0 n = 0 Lemma 5 . Suppose that the assertion about μ ( n ) \mu^{(n)} μ ( n ) (100) at level n n n n + 1 n+1 n + 1 x x x ( n + 1 ) (n+1) ( n + 1 ) ℓ \ell ℓ c = c ℓ ( x ) c=c_\ell(x) c = c ℓ ( x ) x x x ℓ \ell ℓ y y y y − ℓ = x − ℓ y_{-\ell}=x_{-\ell} y − ℓ = x − ℓ y ℓ ≠ x ℓ y_\ell\ne x_\ell y ℓ = x ℓ c c c (100) to each such pre-jump state gives their sum as the measure with density
g v ( n + 1 ) , ℓ ( x , w ) : = ∑ y − ℓ = x − ℓ y ℓ ≠ x ℓ Q ‾ v ℓ ( x ℓ ∣ y ℓ ) f v ( n ) ( y , w ) , v ∈ ( 0 , 1 ) , w ∈ B c , (101) g_v^{(n+1),\ell}(x,w) := \sum_{\substack{y_{-\ell}=x_{-\ell}\\y_\ell\ne x_\ell}} \overline{Q}^{\ell}_v(x^\ell\mid y^\ell)f_v^{(n)}(y,w), \qquad v\in(0,1),\quad w\in\mathcal{B}_c, \tag*{(101)} g v ( n + 1 ) , ℓ ( x , w ) := y − ℓ = x − ℓ y ℓ = x ℓ ∑ Q v ℓ ( x ℓ ∣ y ℓ ) f v ( n ) ( y , w ) , v ∈ ( 0 , 1 ) , w ∈ B c , (101) with respect to d v σ c ( d w ) dv\,\sigma_c(dw) d v σ c ( d w ) φ \varphi φ ( 0 , 1 ) × B c (0,1)\times\mathcal{B}_c ( 0 , 1 ) × B c
E [ 1 { T n + 1 ⋆ ∈ ( 0 , 1 ) , X T n + 1 ⋆ = x , C n + 1 = c } φ ( T n + 1 ⋆ , E T n + 1 ⋆ ) ] = ∫ 0 1 ∫ B c φ ( v , w ) g v ( n + 1 ) , ℓ ( x , w ) σ c ( d w ) d v . (102) \mathbb{E}\left[\mathbf{1}_{\{T^\star_{n+1}\in(0,1),X_{T^\star_{n+1}}=x,C_{n+1}=c\}}\varphi(T^\star_{n+1},E_{T^\star_{n+1}})\right]
=\int_0^1\int_{\mathcal{B}_c}\varphi(v,w)g_v^{(n+1),\ell}(x,w)\,\sigma_c(dw)\,dv. \tag*{(102)} E [ 1 { T n + 1 ⋆ ∈ ( 0 , 1 ) , X T n + 1 ⋆ = x , C n + 1 = c } φ ( T n + 1 ⋆ , E T n + 1 ⋆ ) ] = ∫ 0 1 ∫ B c φ ( v , w ) g v ( n + 1 ) , ℓ ( x , w ) σ c ( d w ) d v . (102) Summing (102) over ℓ \ell ℓ φ = 1 \varphi=1 φ = 1
∑ ℓ = 1 L ∫ 0 1 ∫ B c ℓ ( x ) g v ( n + 1 ) , ℓ ( x , w ) σ c ℓ ( x ) ( d w ) d v = P ( T n + 1 ⋆ ∈ ( 0 , 1 ) , X T n + 1 ⋆ = x ) ≤ 1. (103) \sum_{\ell=1}^{L}\int_0^1\int_{\mathcal{B}_{c_\ell(x)}}g_v^{(n+1),\ell}(x,w)\,\sigma_{c_\ell(x)}(dw)\,dv
=\mathbb{P}(T^\star_{n+1}\in(0,1),X_{T^\star_{n+1}}=x)\le1. \tag*{(103)} ℓ = 1 ∑ L ∫ 0 1 ∫ B c ℓ ( x ) g v ( n + 1 ) , ℓ ( x , w ) σ c ℓ ( x ) ( d w ) d v = P ( T n + 1 ⋆ ∈ ( 0 , 1 ) , X T n + 1 ⋆ = x ) ≤ 1. (103) If λ v ℓ ( x ℓ ) = 0 \lambda_v^\ell(x^\ell)=0 λ v ℓ ( x ℓ ) = 0 Q ‾ v ℓ ( y ℓ ∣ x ℓ ) = 0 \overline{Q}_v^\ell(y^\ell\mid x^\ell)=0 Q v ℓ ( y ℓ ∣ x ℓ ) = 0 y y y (60) and strict positivity of p v ( y ) p_v(y) p v ( y ) Q ‾ v ℓ ( x ℓ ∣ y ℓ ) = 0 \overline{Q}_v^\ell(x^\ell\mid y^\ell)=0 Q v ℓ ( x ℓ ∣ y ℓ ) = 0 (101) is zero at such a time. It satisfies all the hypotheses (84) of Lemma 6.
Given ( T n + 1 ⋆ , X T n + 1 ⋆ , E T n + 1 ⋆ ) = ( v , x , w ) (T^\star_{n+1},X_{T^\star_{n+1}},E_{T^\star_{n+1}})=(v,x,w) ( T n + 1 ⋆ , X T n + 1 ⋆ , E T n + 1 ⋆ ) = ( v , x , w ) w + Δ u ∣ v ( x ) w+\Delta_{u\mid v}(x) w + Δ u ∣ v ( x ) x x x (102) . Lemma 6(i) gives the density of its endpoint at any fixed u < v u<v u < v
g v ( n + 1 ) , ℓ ( x , w ) λ v ℓ ( x ℓ ) when r ∈ R u ℓ ( x ) , v = ρ ℓ ( u , x , r ) , w = r − Δ u ∣ v ( x ) , \frac{g_v^{(n+1),\ell}(x,w)}{\lambda_v^\ell(x^\ell)}
\quad\text{when }r\in R_u^\ell(x),\quad v=\rho_\ell(u,x,r),\quad w=r-\Delta_{u\mid v}(x), λ v ℓ ( x ℓ ) g v ( n + 1 ) , ℓ ( x , w ) when r ∈ R u ℓ ( x ) , v = ρ ℓ ( u , x , r ) , w = r − Δ u ∣ v ( x ) , and zero off the union of these sets. By (101) , this is exactly (97) applied to f ( n ) f^{(n)} f ( n ) H u ( n + 1 ) H_u^{(n+1)} H u ( n + 1 ) (99) follows at level n + 1 n+1 n + 1
Lemma 6(ii) applied to the same entrance measures gives the next-hit formula
Q ‾ v h ( j ∣ x ) f v ( n + 1 ) ( x , w ) d v σ d ( d w ) , d = ( h , j ) , j ≠ x h . \overline{Q}_v^h(j\mid x)f_v^{(n+1)}(x,w)\,dv\,\sigma_d(dw),\qquad d=(h,j),\quad j\ne x^h. Q v h ( j ∣ x ) f v ( n + 1 ) ( x , w ) d v σ d ( d w ) , d = ( h , j ) , j = x h . This is (100) for the ( n + 2 ) (n+2) ( n + 2 ) n n n
Corollary 1. Under the hypotheses of Proposition 1, for every n ≥ 0 n\ge0 n ≥ 0 u ∈ [ 0 , 1 ) u\in[0,1) u ∈ [ 0 , 1 ) x ∈ X x\in\mathcal{X} x ∈ X B ⊆ O B\subseteq\mathcal{O} B ⊆ O
μ u ( n + 1 ) ( { x } × B ) = ∑ ℓ = 1 L ∑ y − ℓ = x − ℓ y ℓ ≠ x ℓ ∫ u 1 ∫ B c ℓ ( x ) 1 { w + Δ u ∣ v ( x ) ∈ B } Q ‾ v ℓ ( x ℓ ∣ y ℓ ) f v ( n ) ( y , w ) σ c ℓ ( x ) ( d w ) d v . (104) \mu_u^{(n+1)}(\{x\}\times B)
=
\sum_{\ell=1}^{L}\ \sum_{\substack{y_{-\ell}=x_{-\ell}\\y^\ell\ne x^\ell}}
\int_u^1\int_{\mathcal{B}_{c_\ell(x)}}
\mathbf{1}_{\{w+\Delta_{u\mid v}(x)\in B\}}
\overline{Q}_v^\ell(x^\ell\mid y^\ell)f_v^{(n)}(y,w)\,\sigma_{c_\ell(x)}(dw)\,dv. \tag*{(104)} μ u ( n + 1 ) ({ x } × B ) = ℓ = 1 ∑ L y − ℓ = x − ℓ y ℓ = x ℓ ∑ ∫ u 1 ∫ B c ℓ ( x ) 1 { w + Δ u ∣ v ( x ) ∈ B } Q v ℓ ( x ℓ ∣ y ℓ ) f v ( n ) ( y , w ) σ c ℓ ( x ) ( d w ) d v . (104) In this integral v = T n + 1 ⋆ v = T_{n+1}^{\star} v = T n + 1 ⋆ u < v < 1 u < v < 1 u < v < 1 r ∈ R u ℓ ( x ) r \in\mathcal{R}^{\ell}_{u}(x) r ∈ R u ℓ ( x ) ( v , w ) (v,w) ( v , w ) (74) ,
f u ( n + 1 ) ( x , r ) F u ( x , r ) = ∑ y − ℓ = x − ℓ y ℓ ≠ x ℓ Q ‾ v ℓ ( y ℓ ∣ x ℓ ) λ v ℓ ( x ℓ ) f v ( n ) ( y , w ) F v ( y , w ) . (105) \frac{f_{u}^{(n+1)}(x,r)}{F_{u}(x,r)} =
\sum_{\substack{y^{-\ell}=x^{-\ell}\\y^{\ell}\ne x^{\ell}}}
\frac{\overline{Q}_{v}^{\ell}(y^{\ell}\mid x^{\ell})}{\lambda_{v}^{\ell}(x^{\ell})}
\frac{f_{v}^{(n)}(y,w)}{F_{v}(y,w)}. \tag*{(105)} F u ( x , r ) f u ( n + 1 ) ( x , r ) = y − ℓ = x − ℓ y ℓ = x ℓ ∑ λ v ℓ ( x ℓ ) Q v ℓ ( y ℓ ∣ x ℓ ) F v ( y , w ) f v ( n ) ( y , w ) . (105) In particular,
f u ( 1 ) ( x , r ) = F u ( x , r ) ∑ y − ℓ = x − ℓ y ℓ ≠ x ℓ Q v ℓ ( y ℓ ∣ x ℓ ) λ v ℓ ( x ℓ ) ∏ h = 1 L 1 { ⟨ y h , w h ⟩ > A v 1 h ( y ) } . (106) f_{u}^{(1)}(x,r) = F_{u}(x,r)
\sum_{\substack{y^{-\ell}=x^{-\ell}\\y^{\ell}\ne x^{\ell}}}
\frac{Q_{v}^{\ell}(y^{\ell}\mid x^{\ell})}{\lambda_{v}^{\ell}(x^{\ell})}
\prod_{h=1}^{L}\mathbf{1}_{\{\langle y^{h},w^{h}\rangle>A_{v_{1}}^{h}(y)\}}. \tag*{(106)} f u ( 1 ) ( x , r ) = F u ( x , r ) y − ℓ = x − ℓ y ℓ = x ℓ ∑ λ v ℓ ( x ℓ ) Q v ℓ ( y ℓ ∣ x ℓ ) h = 1 ∏ L 1 {⟨ y h , w h ⟩ > A v 1 h ( y )} . (106) Off R u ( x ) \mathcal{R}_{u}(x) R u ( x ) f u ( 1 ) ( x , r ) = 0 f_{u}^{(1)}(x,r)=0 f u ( 1 ) ( x , r ) = 0
Proof. Substitute (101) into (86) . Its indicator is equivalent to no further hit before u u u B ⊆ O B\subseteq O B ⊆ O (104) . For (105) , the two identities (60) and (63) give, pointwise at the specified ( u , v , x , y , r , w ) (u,v,x,y,r,w) ( u , v , x , y , r , w )
Q ‾ v ℓ ( x ℓ ∣ y ) F v ( y , w ) = p v ( x ) Q v ℓ ( y ℓ ∣ x ℓ ) γ ( w ) = Q v ℓ ( y ℓ ∣ x ℓ ) F u ( x , r ) . (107) \begin{aligned}
\overline{Q}_{v}^{\ell}(x^{\ell}\mid y)F_{v}(y,w)
= p_{v}(x)Q_{v}^{\ell}(y^{\ell}\mid x^{\ell})\gamma(w) \\
&= Q_{v}^{\ell}(y^{\ell}\mid x^{\ell})F_{u}(x,r). \tag*{(107)}
\end{aligned} Q v ℓ ( x ℓ ∣ y ) F v ( y , w ) = p v ( x ) Q v ℓ ( y ℓ ∣ x ℓ ) γ ( w ) = Q v ℓ ( y ℓ ∣ x ℓ ) F u ( x , r ) . (107) All factors F F F (97) by F u ( x , r ) F_{u}(x,r) F u ( x , r ) (107) in each summand. Finally, substitute (78) at ( v , y , w ) (v,y,w) ( v , y , w ) w ∈ O y w\in\mathcal{O}_{y} w ∈ O y (106) .
Bounds on the partial sums and number of jumps
Lemma 7. Assume Assumption 1. Define F F F (53) on each O x \mathcal{O}_{x} O x f ( 0 ) f^{(0)} f ( 0 ) (78) , and f ( n + 1 ) f^{(n+1)} f ( n + 1 ) (98) . For every u ∈ [ 0 , 1 ) u\in[0,1) u ∈ [ 0 , 1 ) x ∈ X x\in X x ∈ X r ∈ O x r\in\mathcal{O}_{x} r ∈ O x
( J F ) u ( x , r ) = F u ( x , r ) 1 R u ( x ) ( r ) , (108) (\mathcal{J}F)_{u}(x,r)=F_{u}(x,r)\mathbf{1}_{\mathcal{R}_{u}(x)}(r), \tag*{(108)} ( J F ) u ( x , r ) = F u ( x , r ) 1 R u ( x ) ( r ) , (108) f u ( 0 ) ( x , r ) + ( J F ) u ( x , r ) = F u ( x , r ) 1 N u ( x ) ∪ R u ( x ) ( r ) ≤ F u ( x , r ) . (109) f_{u}^{(0)}(x,r)+(\mathcal{J}F)_{u}(x,r)
=F_{u}(x,r)\mathbf{1}_{\mathcal{N}_{u}(x)\cup\mathcal{R}_{u}(x)}(r)
\le F_{u}(x,r). \tag*{(109)} f u ( 0 ) ( x , r ) + ( J F ) u ( x , r ) = F u ( x , r ) 1 N u ( x ) ∪ R u ( x ) ( r ) ≤ F u ( x , r ) . (109) For every integer m ≥ 0 m\ge0 m ≥ 0 ( u , x , r ) (u,x,r) ( u , x , r )
0 ≤ S u ( m ) ( x , r ) : = ∑ n = 0 m f u ( n ) ( x , r ) ≤ F u ( x , r ) . (110) 0\le S_{u}^{(m)}(x,r):=\sum_{n=0}^{m}f_{u}^{(n)}(x,r)\le F_{u}(x,r). \tag*{(110)} 0 ≤ S u ( m ) ( x , r ) := n = 0 ∑ m f u ( n ) ( x , r ) ≤ F u ( x , r ) . (110) These are pointwise statements about the specified functions, including their values at r ∈ B d r\in\mathcal{B}_{d} r ∈ B d d d d x x x
Proof. If r ∉ R u ( x ) r\notin\mathcal{R}_{u}(x) r ∈ / R u ( x ) J \mathcal{J} J (108) . If r ∈ R u ℓ ( x ) r\in\mathcal{R}_{u}^{\ell}(x) r ∈ R u ℓ ( x ) (74) and (61) to calculate
( J F ) u ( x , r ) = γ ( w ) λ v ℓ ( x ℓ ) ∑ y − ℓ = x − ℓ y ℓ ≠ x ℓ p v ( y ) Q v ℓ ( x ℓ ∣ y ) (\mathcal{J}F)_{u}(x,r)
=\frac{\gamma(w)}{\lambda_{v}^{\ell}(x^{\ell})}
\sum_{\substack{y^{-\ell}=x^{-\ell}\\y^{\ell}\ne x^{\ell}}}
p_{v}(y)Q_{v}^{\ell}(x^{\ell}\mid y) ( J F ) u ( x , r ) = λ v ℓ ( x ℓ ) γ ( w ) y − ℓ = x − ℓ y ℓ = x ℓ ∑ p v ( y ) Q v ℓ ( x ℓ ∣ y ) = p v ( x ) γ ( w ) = F u ( x , r ) . =p_{v}(x)\gamma(w)=F_{u}(x,r). = p v ( x ) γ ( w ) = F u ( x , r ) . The last step is (63) . Since N u ( x ) \mathcal{N}_{u}(x) N u ( x ) R u ( x ) \mathcal{R}_{u}(x) R u ( x ) (78) yields (109) at every point of O x \mathcal{O}_{x} O x
For m = 0 m=0 m = 0 (110) follows from (78) . Suppose it holds for an integer m m m J \mathcal{J} J
S u ( m + 1 ) = f u ( 0 ) + J S u ( m ) ≤ f u ( 0 ) + J F ≤ F . S_{u}^{(m+1)}=f_{u}^{(0)}+\mathcal{J}S_{u}^{(m)}
\le f_{u}^{(0)}+\mathcal{J}F\le F. S u ( m + 1 ) = f u ( 0 ) + J S u ( m ) ≤ f u ( 0 ) + J F ≤ F . Every evaluation of S ( m ) S^{(m)} S ( m )
Proposition 2. Assume Assumption 1 and (49) . For u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ] set
M u : = ∫ u 1 ∑ x ∈ X p v ( x ) λ ‾ v ( x ) d v . (111) M_u := \int_u^1 \sum_{x\in\mathcal{X}} p_v(x)\overline{\lambda}_v(x)\,\mathrm{d}v. \tag*{(111)} M u := ∫ u 1 x ∈ X ∑ p v ( x ) λ v ( x ) d v . (111) Then M u < ∞ M_u < \infty M u < ∞ and
M u = ∫ u 1 ∑ x ∈ X p v ( x ) λ v ( x ) d v . (112) M_u = \int_u^1 \sum_{x\in\mathcal{X}} p_v(x)\lambda_v(x)\,\mathrm{d}v. \tag*{(112)} M u = ∫ u 1 x ∈ X ∑ p v ( x ) λ v ( x ) d v . (112) For every u ∈ [ 0 , 1 ) u \in[0,1) u ∈ [ 0 , 1 ) and every integer m ≥ 1 m \ge1 m ≥ 1
∑ q = 1 m P ( T q ⋆ ∈ ( u , 1 ) ) ≤ M u , P ( T m ⋆ ∈ ( u , 1 ) ) ≤ M u m . (113) \sum_{q=1}^{m} \mathbb{P}(T_q^\star\in(u,1)) \le M_u,\qquad\mathbb{P}(T_m^\star\in(u,1)) \le\frac{M_u}{m}. \tag*{(113)} q = 1 ∑ m P ( T q ⋆ ∈ ( u , 1 )) ≤ M u , P ( T m ⋆ ∈ ( u , 1 )) ≤ m M u . (113) Here T q ⋆ T_q^\star T q ⋆ is a successful finite step time from (51) and is set to − ∞ -\infty − ∞ if that step does not exist. The construction almost surely has no failure and only finitely many hits on [ 0 , 1 ] [0,1] [ 0 , 1 ] For every deterministic u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ] its jump count
N u : = ∑ q ≥ 1 1 { T q ⋆ ∈ ( u , 1 ) } (114) N_u := \sum_{q\ge1} \mathbb{1}_{\{T_q^\star\in(u,1)\}} \tag*{(114)} N u := q ≥ 1 ∑ 1 { T q ⋆ ∈ ( u , 1 )} (114) satisfies E N u ≤ M u \mathbb{E}N_u \le M_u E N u ≤ M u
Proof. For each q ∈ { 1 , … , m } q \in\{1,\ldots,m\} q ∈ { 1 , … , m } ψ ( v , w ) = 1 { u < v < 1 } \psi(v,w)=\mathbb{1}_{\{u<v<1\}} ψ ( v , w ) = 1 { u < v < 1 } (100) at level q − 1 q-1 q − 1 q q q
∑ q = 1 m P ( T q ⋆ ∈ ( u , 1 ) ) = ∫ u 1 ∑ x ∈ X ∑ ℓ = 1 L ∑ j : j ≠ x ℓ Q ‾ v ℓ ( j ∣ x ) ∫ B ( ℓ , j ) S v ( m − 1 ) ( x , w ) σ ( ℓ , j ) ( d w ) d v . (115) \sum_{q=1}^{m}\mathbb{P}(T_q^\star\in(u,1))=\int_u^1\sum_{x\in\mathcal{X}}\sum_{\ell=1}^{L}\sum_{j:j\ne x^\ell}\overline{Q}_v^\ell(j\mid x)\int_{B_{(\ell,j)}}S_v^{(m-1)}(x,w)\,\sigma_{(\ell,j)}(\mathrm{d}w)\,\mathrm{d}v. \tag*{(115)} q = 1 ∑ m P ( T q ⋆ ∈ ( u , 1 )) = ∫ u 1 x ∈ X ∑ ℓ = 1 ∑ L j : j = x ℓ ∑ Q v ℓ ( j ∣ x ) ∫ B ( ℓ , j ) S v ( m − 1 ) ( x , w ) σ ( ℓ , j ) ( d w ) d v . (115) Lemma 7 applies pointwise to the functions on these particular faces. For each active d = ( ℓ , j ) d=(\ell,j) d = ( ℓ , j )
∫ B d F v ( x , w ) σ d ( d w ) = p v ( x ) ∫ ( 0 , ∞ ) D − 1 exp ( − ∑ a ≠ d w a ) d w − d = p v ( x ) ∏ a ≠ d ∫ 0 ∞ exp ( − ζ ) d ζ = p v ( x ) . (116) \begin{aligned}
\int_{B_d}F_v(x,w)\,\sigma_d(\mathrm{d}w)=p_v(x)\int_{(0,\infty)^{D-1}}\exp\left(-\sum_{a\ne d}w^a\right)\,\mathrm{d}w^{-d} \\
&=p_v(x)\prod_{a\ne d}\int_0^\infty\exp(-\zeta)\,\mathrm{d}\zeta=p_v(x). \tag*{(116)}
\end{aligned} ∫ B d F v ( x , w ) σ d ( d w ) = p v ( x ) ∫ ( 0 , ∞ ) D − 1 exp − a = d ∑ w a d w − d = p v ( x ) a = d ∏ ∫ 0 ∞ exp ( − ζ ) d ζ = p v ( x ) . (116) Substitution in (115) proves the first inequality in (113) . Lemma 1 gives a common finite bound on λ ‾ v ( x ) \overline{\lambda}_v(x) λ v ( x ) M u < ∞ M_u<\infty M u < ∞
For each fixed v v v (43) gives
∑ x p v ( x ) λ ‾ v ( x ) = ∑ x ∑ y ≠ x p v ( x ) Q ‾ v ( y ∣ x ) = ∑ x ∑ y ≠ x p v ( y ) Q v ( x ∣ y ) = ∑ y p v ( y ) λ v ( y ) . \begin{aligned}
\sum_x p_v(x)\overline{\lambda}_v(x)=\sum_x\sum_{y\ne x}p_v(x)\overline{Q}_v(y\mid x) \\
&=\sum_x\sum_{y\ne x}p_v(y)Q_v(x\mid y)=\sum_y p_v(y)\lambda_v(y).
\end{aligned} x ∑ p v ( x ) λ v ( x ) = x ∑ y = x ∑ p v ( x ) Q v ( y ∣ x ) = x ∑ y = x ∑ p v ( y ) Q v ( x ∣ y ) = y ∑ p v ( y ) λ v ( y ) . Integration proves (112) .
If T m ⋆ ∈ ( u , 1 ) T_m^\star\in(u,1) T m ⋆ ∈ ( u , 1 ) T q ⋆ ∈ ( u , 1 ) T_q^\star\in(u,1) T q ⋆ ∈ ( u , 1 ) 1 ≤ q ≤ m 1\le q\le m 1 ≤ q ≤ m
m P ( T m ⋆ ∈ ( u , 1 ) ) ≤ ∑ q = 1 m P ( T q ⋆ ∈ ( u , 1 ) ) ≤ M u , m\mathbb{P}(T_m^\star\in(u,1))\le\sum_{q=1}^{m}\mathbb{P}(T_q^\star\in(u,1))\le M_u, m P ( T m ⋆ ∈ ( u , 1 )) ≤ q = 1 ∑ m P ( T q ⋆ ∈ ( u , 1 )) ≤ M u , which proves the second inequality.
For each finite index, Proposition 1 assigns probability zero to failure of the hitting conditions, including a hit at 0. The union of these failure events over the positive integers is null. On its complement, an infinite number of hits in [ 0 , 1 ] [0,1] [ 0 , 1 ] T n ⋆ ∈ ( 0 , 1 ) T_n^{\star} \in(0,1) T n ⋆ ∈ ( 0 , 1 ) n n n n n n (113) at u = 0 u=0 u = 0
P ( T n ⋆ ∈ ( 0 , 1 ) for all n ≥ 1 ) = lim n → ∞ P ( T n ⋆ ∈ ( 0 , 1 ) ) = 0. \mathbb{P}\left(T_n^{\star} \in(0,1)\ \text{for all }n \ge1\right)=\lim_{n\to\infty}\mathbb{P}\left(T_n^{\star} \in(0,1)\right)=0. P ( T n ⋆ ∈ ( 0 , 1 ) for all n ≥ 1 ) = n → ∞ lim P ( T n ⋆ ∈ ( 0 , 1 ) ) = 0. Thus a successful finite construction eventually reaches a segment having no further hit down to 0. This yields a path on all of [ 0 , 1 ] [0,1] [ 0 , 1 ]
Finally, monotone convergence applied to the first inequality in (113) gives
E N u = ∑ q ≥ 1 P ( T q ⋆ ∈ ( u , 1 ) ) ≤ M u . \mathbb{E}N_u=\sum_{q\ge1}\mathbb{P}\left(T_q^{\star}\in(u,1)\right)\le M_u. E N u = q ≥ 1 ∑ P ( T q ⋆ ∈ ( u , 1 ) ) ≤ M u . At u = 1 u=1 u = 1
Proof of Theorem I.1
Proof of Theorem I.1. Proposition 2 gives the almost-sure construction, finiteness of the number of jumps, and absence of a hit at every prescribed deterministic time.
Fix u ∈ [ 0 , 1 ) u\in[0,1) u ∈ [ 0 , 1 ) u u u H u ( n ) H_u^{(n)} H u ( n ) n ≥ 0 n\ge0 n ≥ 0 H u ( n ) = { N u = n } H_u^{(n)}=\{N_u=n\} H u ( n ) = { N u = n } (56) and Proposition 1, the law at time u u u X × O \mathcal{X}\times\mathcal{O} X × O
f u ( x , r ) : = ∑ n = 0 ∞ f u ( n ) ( x , r ) , x ∈ X , r ∈ O . (117) f_u(x,r):=\sum_{n=0}^{\infty}f_u^{(n)}(x,r),\qquad x\in\mathcal{X},\quad r\in\mathcal{O}. \tag*{(117)} f u ( x , r ) := n = 0 ∑ ∞ f u ( n ) ( x , r ) , x ∈ X , r ∈ O . (117) Lemma 7 and monotone convergence give 0 ≤ f u ( x , r ) ≤ F u ( x , r ) 0\le f_u(x,r)\le F_u(x,r) 0 ≤ f u ( x , r ) ≤ F u ( x , r ) H u ( n ) H_u^{(n)} H u ( n )
∑ x ∫ O f u ( x , r ) d r = ∑ n = 0 ∞ ∑ x ∫ O f u ( n ) ( x , r ) d r = ∑ n = 0 ∞ P ( H u ( n ) ) = 1. \sum_{x}\int_{\mathcal{O}}f_u(x,r)\,\mathrm{d}r
=\sum_{n=0}^{\infty}\sum_x\int_{\mathcal{O}}f_u^{(n)}(x,r)\,\mathrm{d}r
=\sum_{n=0}^{\infty}\mathbb{P}\left(H_u^{(n)}\right)=1. x ∑ ∫ O f u ( x , r ) d r = n = 0 ∑ ∞ x ∑ ∫ O f u ( n ) ( x , r ) d r = n = 0 ∑ ∞ P ( H u ( n ) ) = 1. By the definition of F u F_u F u
∑ x ∫ O F u ( x , r ) d r = ∑ x p u ( x ) ∏ c ∈ [ 1 , L ] × [ 1 , K ] ∫ 0 ∞ exp ( − ζ ) d ζ = 1. \sum_x\int_{\mathcal{O}}F_u(x,r)\,\mathrm{d}r
=\sum_x p_u(x)\prod_{c\in[1,L]\times[1,K]}\int_0^\infty\exp(-\zeta)\,\mathrm{d}\zeta=1. x ∑ ∫ O F u ( x , r ) d r = x ∑ p u ( x ) c ∈ [ 1 , L ] × [ 1 , K ] ∏ ∫ 0 ∞ exp ( − ζ ) d ζ = 1. Consequently,
∑ x ∫ O ( F u ( x , r ) − f u ( x , r ) ) d r = 0. \sum_x\int_{\mathcal{O}}\left(F_u(x,r)-f_u(x,r)\right)\,\mathrm{d}r=0. x ∑ ∫ O ( F u ( x , r ) − f u ( x , r ) ) d r = 0. Its integrand is nonnegative, so f u ( x , ⋅ ) = F u ( x , ⋅ ) f_u(x,\cdot)=F_u(x,\cdot) f u ( x , ⋅ ) = F u ( x , ⋅ ) r r r x x x (54) for every deterministic u < 1 u<1 u < 1 u = 1 u=1 u = 1 (49) . Integrating the product density over the coordinate box 0 < r ℓ j ≤ ν ℓ j 0<r^{\ell j}\le\nu^{\ell j} 0 < r ℓ j ≤ ν ℓ j
p u ( x ) ∏ ( ℓ , j ) ∈ [ 1 , L ] × [ 1 , K ] ∫ 0 ν ℓ j exp ( − ζ ) d ζ = p u ( x ) ∏ ( ℓ , j ) ∈ [ 1 , L ] × [ 1 , K ] ( 1 − exp ( − ν ℓ j ) ) , p_u(x)\prod_{(\ell,j)\in[1,L]\times[1,K]}\int_0^{\nu^{\ell j}}\exp(-\zeta)\,\mathrm{d}\zeta
=p_u(x)\prod_{(\ell,j)\in[1,L]\times[1,K]}\left(1-\exp(-\nu^{\ell j})\right), p u ( x ) ( ℓ , j ) ∈ [ 1 , L ] × [ 1 , K ] ∏ ∫ 0 ν ℓ j exp ( − ζ ) d ζ = p u ( x ) ( ℓ , j ) ∈ [ 1 , L ] × [ 1 , K ] ∏ ( 1 − exp ( − ν ℓ j ) ) , which gives the corresponding distribution function.
Corollary 2. Under the hypotheses of Theorem I.1, for every deterministic u ∈ [ 0 , 1 ) u \in[0,1) u ∈ [ 0 , 1 ) m ≥ 0 m \ge0 m ≥ 0
∑ x ∫ O ( F u ( x , r ) − ∑ n = 0 m f u ( n ) ( x , r ) ) d r = P ( N u ≥ m + 1 ) ≤ M u m + 1 . (118) \sum_x \int_{\mathcal{O}} \left(F_u(x,\mathbf{r})-\sum_{n=0}^{m}f_u^{(n)}(x,\mathbf{r})\right)\,d\mathbf{r}=\mathbb{P}(N_u\ge m+1)\le\frac{M_u}{m+1}. \tag*{(118)} x ∑ ∫ O ( F u ( x , r ) − n = 0 ∑ m f u ( n ) ( x , r ) ) d r = P ( N u ≥ m + 1 ) ≤ m + 1 M u . (118) Proof. The integrand is nonnegative by (110) . Subtract the masses of the events H u ( 0 ) , … , H u ( m ) H_u^{(0)},\ldots,H_u^{(m)} H u ( 0 ) , … , H u ( m ) (117) . This gives the equality. Except on a null set, { N u ≥ m + 1 } = { T m + 1 ∈ ( u , 1 ) } \{N_u\ge m+1\}=\{T_{m+1}\in(u,1)\} { N u ≥ m + 1 } = { T m + 1 ∈ ( u , 1 )} (113) .
Last-jump distribution
Proposition 3. Assume the hypotheses of Theorem I.1. Fix u ∈ [ 0 , 1 ) u\in[0,1) u ∈ [ 0 , 1 ) { N u ≥ 1 } \{N_u\ge1\} { N u ≥ 1 }
R u : = T N u ∗ , I u : = the first coordinate of C N u . R_u:=T_{N_u}^{\ast},\qquad I_u:=\text{the first coordinate of }C_{N_u}. R u := T N u ∗ , I u := the first coordinate of C N u . For every x ∈ X x\in X x ∈ X ℓ ∈ [ 1 , L ] \ell\in[1,L] ℓ ∈ [ 1 , L ] c = c ℓ ( x ) c=c_\ell(x) c = c ℓ ( x ) ( R u , E u − c ) (R_u,E_u^{-c}) ( R u , E u − c ) { N u ≥ 1 , X u = x , I u = ℓ } \{N_u\ge1,X_u=x,I_u=\ell\} { N u ≥ 1 , X u = x , I u = ℓ }
P ( N u ≥ 1 , X u = x , I u = ℓ , R u ∈ d v , E u − c ∈ d r − c ) = p u ( x ) λ v ℓ ( x ) exp [ − A u ∣ v ℓ ( x ) − ∑ a ≠ c r a ] ∏ h ≠ ℓ 1 { ⟨ x h , r h ⟩ > A u ∣ v h ( x ) } d v d r − c , (119) \mathbb{P}(N_u\ge1,X_u=x,I_u=\ell,R_u\in dv,E_u^{-c}\in d\mathbf{r}^{-c})
=p_u(x)\lambda_v^\ell(x)\exp\left[-A_{u|v}^\ell(x)-\sum_{a\ne c}r^a\right]\prod_{h\ne\ell}\mathbf{1}_{\{\langle x^h,r^h\rangle>A_{u|v}^h(x)\}}\,dv\,d\mathbf{r}^{-c}, \tag*{(119)} P ( N u ≥ 1 , X u = x , I u = ℓ , R u ∈ d v , E u − c ∈ d r − c ) = p u ( x ) λ v ℓ ( x ) exp − A u ∣ v ℓ ( x ) − a = c ∑ r a h = ℓ ∏ 1 {⟨ x h , r h ⟩ > A u ∣ v h ( x )} d v d r − c , (119) on ( u , 1 ) × ( 0 , ∞ ) D − 1 (u,1)\times(0,\infty)^{D-1} ( u , 1 ) × ( 0 , ∞ ) D − 1 E u c = A u ∣ R u ℓ ( x ) E_u^c=A_{u|R_u}^\ell(x) E u c = A u ∣ R u ℓ ( x ) (78) .
Proof. Almost surely there are finitely many successful jumps, no jump at u u u { N u ≥ 1 , X u = x } \{N_u\ge1,X_u=x\} { N u ≥ 1 , X u = x } v = R u v=R_u v = R u u u u E u ∈ R I u ( x ) E_u\in R^{I_u}(x) E u ∈ R I u ( x ) R u = ρ I u ( u , x , E u ) R_u=\rho_{I_u}(u,x,E_u) R u = ρ I u ( u , x , E u ) E u c I u ( x ) = A u ∣ R u I u ( x ) E_u^{c_{I_u}(x)}=A_{u|R_u}^{I_u}(x) E u c I u ( x ) = A u ∣ R u I u ( x ) N u ( x ) \mathcal{N}_u(x) N u ( x )
Theorem I.1 supplies the interior density p u ( x ) γ ( r ) p_u(x)\gamma(\mathbf{r}) p u ( x ) γ ( r ) R u ℓ ( x ) ∩ O R_u^\ell(x)\cap\mathcal{O} R u ℓ ( x ) ∩ O r c r^c r c v v v
r c = A u ∣ v ℓ ( x ) , r a = r a ( a ≠ c ) . r^c=A_{u|v}^\ell(x),\qquad r^a=r^a\quad(a\ne c). r c = A u ∣ v ℓ ( x ) , r a = r a ( a = c ) . The determinant in these coordinates is λ v ℓ ( x ) \lambda_v^\ell(x) λ v ℓ ( x ) ℓ \ell ℓ
⟨ x h , r h ⟩ > A u ∣ v h ( x ) ( h ≠ ℓ ) . \langle x^h,r^h\rangle>A_{u|v}^h(x)\quad(h\ne\ell). ⟨ x h , r h ⟩ > A u ∣ v h ( x ) ( h = ℓ ) . All active coordinates are already positive. On the set λ v ℓ ( x ℓ ) > 0 \lambda_v^\ell(x^\ell)>0 λ v ℓ ( x ℓ ) > 0 r c r^c r c (119) is zero where λ v ℓ ( x ℓ ) = 0 \lambda_v^\ell(x^\ell)=0 λ v ℓ ( x ℓ ) = 0 r c = A u ∣ v ℓ ( x ) r^c=A_{u|v}^\ell(x) r c = A u ∣ v ℓ ( x ) (119) .
Corollary 3. Under the hypotheses of Theorem I.1, suppose L = 1 L=1 L = 1 V V V u ∈ [ 0 , 1 ) u\in[0,1) u ∈ [ 0 , 1 ) j ∈ [ 1 , K ] j\in[1,K] j ∈ [ 1 , K ] r ∈ O j \mathbf{r}\in\mathcal{O}_j r ∈ O j
f u ( 0 ) ( j , r ) = p u ( j ) γ ( r ) 1 { r 1 j > ∫ u 1 λ τ 1 ( j ) d τ } . f_u^{(0)}(j,\mathbf{r})=p_u(j)\gamma(\mathbf{r})\mathbf{1}_{\{r^{1j}>\int_u^1\lambda_\tau^1(j)\,d\tau\}}. f u ( 0 ) ( j , r ) = p u ( j ) γ ( r ) 1 { r 1 j > ∫ u 1 λ τ 1 ( j ) d τ } . If r ∈ R u ( j ) \mathbf{r}\in R_u(j) r ∈ R u ( j ) v ∈ ( u , 1 ) v\in(u,1) v ∈ ( u , 1 ) λ v 1 ( j ) > 0 \lambda_v^1(j)>0 λ v 1 ( j ) > 0
r 1 j = ∫ u v λ τ 1 ( j ) d τ , r^{1j}=\int_u^v\lambda_\tau^1(j)\,d\tau, r 1 j = ∫ u v λ τ 1 ( j ) d τ , and set w 1 j = 0 w^{1j}=0 w 1 j = 0 w 1 k = r 1 k + ∫ u v Q ‾ τ 1 ( k ∣ j ) d τ w^{1k}=r^{1k}+\int_u^v\overline{Q}_\tau^1(k\mid j)\,d\tau w 1 k = r 1 k + ∫ u v Q τ 1 ( k ∣ j ) d τ k ≠ j k\ne j k = j n ≥ 0 n\ge0 n ≥ 0
f u ( n + 1 ) ( j , r ) = 1 λ v 1 ( j ) ∑ k ∈ [ 1 , K ] ∖ { j } Q ‾ v 1 ( j ∣ k ) f v ( n ) ( k , w ) . (120) f_u^{(n+1)}(j,\mathbf{r})=\frac{1}{\lambda_v^1(j)}\sum_{k\in[1,K]\setminus\{j\}}\overline{Q}_v^1(j\mid k)f_v^{(n)}(k,\mathbf{w}). \tag*{(120)} f u ( n + 1 ) ( j , r ) = λ v 1 ( j ) 1 k ∈ [ 1 , K ] ∖ { j } ∑ Q v 1 ( j ∣ k ) f v ( n ) ( k , w ) . (120) It is zero when r ∉ R u ( j ) r \notin\mathcal{R}_u(j) r ∈ / R u ( j )
Proof. There is one passive residual, namely c 1 ( j ) = ( 1 , j ) c_1(j) = (1,j) c 1 ( j ) = ( 1 , j ) (70) reduces to the single inequality in the no-jump formula. In (97) , the only possible value of ℓ \ell ℓ k k k j j j Q ‾ 1 ( j ∣ k ) \overline{Q}_1(j\mathbin{|}k) Q 1 ( j ∣ k ) (58) and (74) give the stated values of w w w (120) and the zero convention directly from the general recursion.
