Introduction and main results
We develop a general approach to establish quantitative convergence in Rényi divergence of order two, and apply it to three kinetic samplers: underdamped Langevin dynamics (ULD) [12 ] , the bouncy particle sampler (BPS) [10 ] , and the Zigzag sampler (ZZ) [6 ] . We assume that the target measure satisfies a logarithmic Sobolev inequality (LSI) and that its potential has a finite Hessian lower bound, which may be negative. ZZ also requires an upper Hessian bound, which BPS uses when it is available. For an initial law whose Rényi-2 divergence from equilibrium is at most R R R R R R R R R R log R \sqrt{R\log R} R log R
Hypocoercivity estimates give exponential convergence in L 2 L^2 L 2 [3 , 11 , 20 , 29 ] . Converted directly, such estimates give mixing times linear in the initial divergence, because the
2020 Mathematics Subject Classification. Primary 65C05; Secondary 60J25, 35Q84, 35B40, 47D07.
Key words and phrases. Hypocoercivity, Rényi divergence, kinetic sampling, logarithmic Sobolev inequality, bouncy particle sampler, Zigzag process.
This work was partially supported by the National Science Foundation under grant DMS-2309378. We thank Sinho Chewi for helpful discussions. χ 2 \chi^2 χ 2 R log R \sqrt{R\log R} R log R d ≥ 2 d \ge2 d ≥ 2
Throughout, we assume U ∈ C 2 ( R d ) U \in C^2(\mathbb{R}^d) U ∈ C 2 ( R d ) U ( x ) → ∞ U(x) \to\infty U ( x ) → ∞ ∣ x ∣ → ∞ |x| \to\infty ∣ x ∣ → ∞ ∫ e − U d x < ∞ \int e^{-U}\,dx < \infty ∫ e − U d x < ∞ α \alpha α β \beta β
α I d ⪯ ∇ 2 U ( x ) ⪯ β I d , x ∈ R d . (1) \alpha I_d \preceq\nabla^2 U(x) \preceq\beta I_d,\qquad x \in\mathbb{R}^d. \tag*{(1)} α I d ⪯ ∇ 2 U ( x ) ⪯ β I d , x ∈ R d . (1) The lower bound α \alpha α β = + ∞ \beta= +\infty β = + ∞ z + = max { z , 0 } z_+ = \max\{z,0\} z + = max { z , 0 } ( − α ) + (-\alpha)_+ ( − α ) +
The position target and the joint equilibrium measure are
μ x ( d x ) ∝ e − U ( x ) d x , μ = μ x ⊗ μ v , μ v = N ( 0 , I d ) . \mu_x(dx) \propto e^{-U(x)}\,dx,\qquad\mu= \mu_x \otimes\mu_v,\qquad\mu_v = N(0,I_d). μ x ( d x ) ∝ e − U ( x ) d x , μ = μ x ⊗ μ v , μ v = N ( 0 , I d ) . We assume that μ x \mu_x μ x C L S I > 0 C_{\mathrm{LSI}} > 0 C LSI > 0
Ent μ x ( h 2 ) ≤ 2 C L S I ∥ ∇ x h ∥ L 2 ( μ x ) 2 . (2) \operatorname{Ent}_{\mu_x}(h^2) \le\frac{2}{C_{\mathrm{LSI}}}\|\nabla_x h\|_{L^2(\mu_x)}^2. \tag*{(2)} Ent μ x ( h 2 ) ≤ C LSI 2 ∥ ∇ x h ∥ L 2 ( μ x ) 2 . (2) Here Ent μ x ( h ) = ∫ h log ( h / ∫ h d μ x ) d μ x \operatorname{Ent}_{\mu_x}(h) = \int h\log(h/\int h\,d\mu_x)\,d\mu_x Ent μ x ( h ) = ∫ h log ( h / ∫ h d μ x ) d μ x h h h C L S I C_{\mathrm{LSI}} C LSI α > 0 \alpha> 0 α > 0 C L S I = α C_{\mathrm{LSI}} = \alpha C LSI = α [4 ] ; see also [5 Proposition 5.7.1] . We keep C L S I C_{\mathrm{LSI}} C LSI α \alpha α
ULD couples Hamiltonian transport to friction and noise in the velocity. BPS and ZZ are piecewise deterministic Markov processes (PDMPs) whose position moves at constant velocity between velocity jumps: bounces for BPS, coordinate flips for ZZ, and refreshments for both. Let Π v h ( x ) = ∫ h ( x , v ) d μ v ( v ) \Pi_v h(x) = \int h(x,v)\,d\mu_v(v) Π v h ( x ) = ∫ h ( x , v ) d μ v ( v ) γ > 0 \gamma> 0 γ > 0 h ∈ C c ∞ ( R 2 d ) h \in C_c^\infty(\mathbb{R}^{2d}) h ∈ C c ∞ ( R 2 d )
L h = ( v ⋅ ∇ x − ∇ U ⋅ ∇ v ) h + γ ( Δ v h − v ⋅ ∇ v h ) (3) \mathcal{L}h = (v \cdot\nabla_x - \nabla U \cdot\nabla_v)h + \gamma(\Delta_v h - v \cdot\nabla_v h) \tag*{(3)} L h = ( v ⋅ ∇ x − ∇ U ⋅ ∇ v ) h + γ ( Δ v h − v ⋅ ∇ v h ) (3) (ULD),
L h = v ⋅ ∇ x h + ( v ⋅ ∇ U ) + [ h ( x , R x v ) − h ( x , v ) ] + γ ( Π v h − h ) (4) \mathcal{L}h = v \cdot\nabla_x h + (v \cdot\nabla U)_+[h(x,R_xv) - h(x,v)] + \gamma(\Pi_v h - h) \tag*{(4)} L h = v ⋅ ∇ x h + ( v ⋅ ∇ U ) + [ h ( x , R x v ) − h ( x , v )] + γ ( Π v h − h ) (4) (BPS),
L h = v ⋅ ∇ x h + ∑ i = 1 d ( v i ∂ x i U ) + [ h ( x , F i v ) − h ( x , v ) ] + γ ( Π v h − h ) (5) \mathcal{L}h = v \cdot\nabla_x h + \sum_{i=1}^{d}(v_i\partial_{x_i}U)_+[h(x,F_i v) - h(x,v)] + \gamma(\Pi_v h - h) \tag*{(5)} L h = v ⋅ ∇ x h + i = 1 ∑ d ( v i ∂ x i U ) + [ h ( x , F i v ) − h ( x , v )] + γ ( Π v h − h ) (5) (ZZ).
Here R x v = v − 2 ( v ⋅ n ) n R_xv = v - 2(v \cdot n)n R x v = v − 2 ( v ⋅ n ) n v v v n = ∇ U / ∣ ∇ U ∣ n = \nabla U/|\nabla U| n = ∇ U /∣∇ U ∣ U U U n n n F i v = v − 2 v i e i F_i v = v - 2v_i e_i F i v = v − 2 v i e i i i i μ v \mu_v μ v γ \gamma γ
Under the assumptions above, the three dynamics are well posed and preserve μ \mu μ P t = e t L \mathcal{P}_t = e^{t\mathcal{L}} P t = e t L ∗ * ∗ L 2 ( μ ) L^2(\mu) L 2 ( μ ) P t \mathcal{P}_t P t P t ∗ \mathcal{P}_t^* P t ∗ L 2 ( μ ) L^2(\mu) L 2 ( μ ) μ \mu μ ρ t \rho_t ρ t t t t
For a probability measure ϱ ≪ μ \varrho\ll\mu ϱ ≪ μ
R 2 ( ϱ ∣ μ ) = log ( ∫ ( d ϱ d μ ) 2 d μ ) = log ( 1 + χ 2 ( ϱ ∣ μ ) ) , (6) R_2(\varrho\mid\mu)=\log\left(\int\left(\frac{\mathrm{d}\varrho}{\mathrm{d}\mu}\right)^2\mathrm{d}\mu\right)=\log(1+\chi^2(\varrho\mid\mu)), \tag*{(6)} R 2 ( ϱ ∣ μ ) = log ( ∫ ( d μ d ϱ ) 2 d μ ) = log ( 1 + χ 2 ( ϱ ∣ μ )) , (6) where χ 2 ( ϱ ∣ μ ) = ∥ d ϱ d μ − 1 ∥ L 2 ( μ ) 2 \chi^2(\varrho\mid\mu)=\left\|\frac{\mathrm{d}\varrho}{\mathrm{d}\mu}-1\right\|_{L^2(\mu)}^2 χ 2 ( ϱ ∣ μ ) = d μ d ϱ − 1 L 2 ( μ ) 2 χ 2 \chi^2 χ 2
TV ( ϱ , μ ) = 1 2 ∫ ∣ d ϱ d μ − 1 ∣ d μ . \operatorname{TV}(\varrho,\mu)=\frac{1}{2}\int\left|\frac{\mathrm{d}\varrho}{\mathrm{d}\mu}-1\right|\mathrm{d}\mu. TV ( ϱ , μ ) = 2 1 ∫ d μ d ϱ − 1 d μ . By Jensen’s inequality, R 2 R_2 R 2
TV ( ϱ , μ ) 2 ≤ 1 2 R 2 ( ϱ ∣ μ ) . (7) \operatorname{TV}(\varrho,\mu)^2\leq\frac{1}{2}R_2(\varrho\mid\mu). \tag*{(7)} TV ( ϱ , μ ) 2 ≤ 2 1 R 2 ( ϱ ∣ μ ) . (7) Theorem 1.1 (Exponential decay of Rényi-2 divergence). Assume (2) and (1) with α ∈ R \alpha\in\mathbb{R} α ∈ R where β = + ∞ \beta=+\infty β = + ∞ is allowed for ULD and BPS and β < ∞ \beta<\infty β < ∞ is required for ZZ. Let ϱ t \varrho_t ϱ t be the law at time t t t of ULD, BPS, or ZZ with friction or refreshment rate γ > 0 \gamma>0 γ > 0 and let R ≥ 0 R\geq0 R ≥ 0 satisfy R 2 ( ϱ 0 ∣ μ ) ≤ R R_2(\varrho_0\mid\mu)\leq R R 2 ( ϱ 0 ∣ μ ) ≤ R Then, for every t ≥ 0 t\geq0 t ≥ 0
R 2 ( ϱ t ∣ μ ) ≤ 3 e − Λ t R 2 ( ϱ 0 ∣ μ ) , (8) R_2(\varrho_t\mid\mu)\leq3e^{-\Lambda t}R_2(\varrho_0\mid\mu), \tag*{(8)} R 2 ( ϱ t ∣ μ ) ≤ 3 e − Λ t R 2 ( ϱ 0 ∣ μ ) , (8) where
Λ = 2 C L S I γ 9 [ γ 2 + ( − α ) + + 3 C L S I ( 1 + R ) ] (9) \Lambda=\frac{2C_{\mathrm{LSI}}\gamma}{9[\gamma^2+(-\alpha)_++3C_{\mathrm{LSI}}(1+R)]} \tag*{(9)} Λ = 9 [ γ 2 + ( − α ) + + 3 C LSI ( 1 + R )] 2 C LSI γ (9) for ULD;
Λ = 2 C L S I γ 900 [ γ 2 + min { d [ C L S I ( 1 + R ) + ( − α ) + ] , C L S I ( 1 + R ) + ( − α ) + + β d } ] (10) \Lambda=\frac{2C_{\mathrm{LSI}}\gamma}{900[\gamma^2+\min\{d[C_{\mathrm{LSI}}(1+R)+(-\alpha)_+],C_{\mathrm{LSI}}(1+R)+(-\alpha)_++\beta d\}]} \tag*{(10)} Λ = 900 [ γ 2 + min { d [ C LSI ( 1 + R ) + ( − α ) + ] , C LSI ( 1 + R ) + ( − α ) + + β d }] 2 C LSI γ (10) for BPS;
Λ = 2 C L S I γ 900 [ γ 2 + β + ( − α ) + + C L S I ( 1 + R ) ] (11) \Lambda=\frac{2C_{\mathrm{LSI}}\gamma}{900[\gamma^2+\beta+(-\alpha)_++C_{\mathrm{LSI}}(1+R)]} \tag*{(11)} Λ = 900 [ γ 2 + β + ( − α ) + + C LSI ( 1 + R )] 2 C LSI γ (11) for ZZ.
The LSI is used only in the nonlinear spatial resolvent estimate, Proposition 3.4 , and negative curvature enters only through ( − α ) + (-\alpha)_+ ( − α ) + α ≥ 0 \alpha\geq0 α ≥ 0 (10) selects the better of two bounds for the BPS cross term. One uses only the lower Hessian bound and produces the β \beta β d [ C L S I ( 1 + R ) + ( − α ) + ] d[C_{\mathrm{LSI}}(1+R)+(-\alpha)_+] d [ C LSI ( 1 + R ) + ( − α ) + ] C L S I ( 1 + R ) + ( − α ) + + β d C_{\mathrm{LSI}}(1+R)+(-\alpha)_++\beta d C LSI ( 1 + R ) + ( − α ) + + β d β = + ∞ \beta=+\infty β = + ∞ d [ C L S I ( 1 + R ) + ( − α ) + ] d[C_{\mathrm{LSI}}(1+R)+(-\alpha)_+] d [ C LSI ( 1 + R ) + ( − α ) + ]
For δ > 0 \delta>0 δ > 0
T δ T V = inf { t ≥ 0 : TV ( ϱ t , μ ) ≤ δ } . T_\delta^{\mathrm{TV}}=\inf\{t\geq0:\operatorname{TV}(\varrho_t,\mu)\leq\delta\}. T δ TV = inf { t ≥ 0 : TV ( ϱ t , μ ) ≤ δ } . Corollary 1.2 (Rényi-2 and total variation mixing). Under the assumptions of Theorem 1.1 , let R ≥ 1 R\geq1 R ≥ 1 satisfy R 2 ( ϱ 0 ∣ μ ) ≤ R R_2(\varrho_0\mid\mu)\leq R R 2 ( ϱ 0 ∣ μ ) ≤ R and let 0 < η ≤ 1 / 2 0<\eta\leq1/2 0 < η ≤ 1/2 Write L = log ( 3 R / η ) \mathcal{L}=\log(3R/\eta) L = log ( 3 R / η ) and choose
γ 2 = { 3 C L S I ( 1 + R / L ) + ( − α ) + , ULD , min { d [ C L S I ( 1 + R / L ) + ( − α ) + ] , C L S I ( 1 + R / L ) + ( − α ) + + β d } , BPS , C L S I ( 1 + R / L ) + β + ( − α ) + , ZZ . (12) \gamma^2=
\begin{cases}
3C_{\mathrm{LSI}}(1+R/\mathcal{L})+(-\alpha)_+, & \text{ULD},\\
\min\{d[C_{\mathrm{LSI}}(1+R/\mathcal{L})+(-\alpha)_+],C_{\mathrm{LSI}}(1+R/\mathcal{L})+(-\alpha)_++\beta d\}, & \text{BPS},\\
C_{\mathrm{LSI}}(1+R/\mathcal{L})+\beta+(-\alpha)_+, & \text{ZZ}.
\end{cases} \tag*{(12)} γ 2 = ⎩ ⎨ ⎧ 3 C LSI ( 1 + R / L ) + ( − α ) + , min { d [ C LSI ( 1 + R / L ) + ( − α ) + ] , C LSI ( 1 + R / L ) + ( − α ) + + β d } , C LSI ( 1 + R / L ) + β + ( − α ) + , ULD , BPS , ZZ . (12) Then R 2 ( ϱ t ∣ μ ) ≤ η R_2(\varrho_t \mid\mu) \le\eta R 2 ( ϱ t ∣ μ ) ≤ η t t t
t ≥ L C L S I { 60 3 ( 1 + R / L ) + ( − α ) + / C L S I , U L D , 6000 min { d [ 1 + R / L + ( − α ) + / C L S I ] , 1 + R / L + [ ( − α ) + + β d ] / C L S I } , B P S , 6000 1 + R / L + [ β + ( − α ) + ] / C L S I , Z Z . (13) t \ge\frac{\mathcal{L}}{\sqrt{C_{\mathrm{LSI}}}}
\begin{cases}
60\sqrt{3(1+R/\mathcal{L})+(-\alpha)_+/C_{\mathrm{LSI}}}, & \mathrm{ULD},\\
6000\sqrt{\min\{d[1+R/\mathcal{L}+(-\alpha)_+/C_{\mathrm{LSI}}],\,1+R/\mathcal{L}+[(-\alpha)_+ + \beta d]/C_{\mathrm{LSI}}\}}, & \mathrm{BPS},\\
6000\sqrt{1+R/\mathcal{L}+[\beta+(-\alpha)_+]/C_{\mathrm{LSI}}}, & \mathrm{ZZ}.
\end{cases} \tag*{(13)} t ≥ C LSI L ⎩ ⎨ ⎧ 60 3 ( 1 + R / L ) + ( − α ) + / C LSI , 6000 min { d [ 1 + R / L + ( − α ) + / C LSI ] , 1 + R / L + [( − α ) + + β d ] / C LSI } , 6000 1 + R / L + [ β + ( − α ) + ] / C LSI , ULD , BPS , ZZ . (13) For 0 < δ ≤ 1 / 2 0<\delta\le1/2 0 < δ ≤ 1/2 η = 2 δ 2 \eta=2\delta^2 η = 2 δ 2 L = log [ 3 R / ( 2 δ 2 ) ] \mathcal{L}=\log[3R/(2\delta^2)] L = log [ 3 R / ( 2 δ 2 )] T δ T V T_\delta^{\mathrm{TV}} T δ TV (13) .
The total variation bound follows from the Rényi-2 bound by (7) . By data processing, both bounds also hold for the position marginal. The numerical constants are not optimized.
The initial divergence can be large in high dimension. Suppose that 0 < α ≤ β < ∞ 0<\alpha\le\beta<\infty 0 < α ≤ β < ∞ x ⋆ x_\star x ⋆ U U U ϱ 0 = N ( x ⋆ , β − 1 I d ) ⊗ μ v \varrho_0=N(x_\star,\beta^{-1}I_d)\otimes\mu_v ϱ 0 = N ( x ⋆ , β − 1 I d ) ⊗ μ v μ x \mu_x μ x ( β / α ) d / 2 (\beta/\alpha)^{d/2} ( β / α ) d /2 [18 ] , so R 2 ( ϱ 0 ∣ μ ) ≤ d 2 log ( β / α ) R_2(\varrho_0\mid\mu)\le\frac{d}{2}\log(\beta/\alpha) R 2 ( ϱ 0 ∣ μ ) ≤ 2 d log ( β / α ) Table 1 compares the bounds of Corollary 1.2 for R ≍ d R\asymp d R ≍ d L 2 L^2 L 2 U U U R 2 ( ϱ 0 ∣ μ ) ≤ R R_2(\varrho_0\mid\mu)\le R R 2 ( ϱ 0 ∣ μ ) ≤ R α < 0 \alpha<0 α < 0 α \alpha α § 1.1 discusses these methods and compares the resulting gradient-query complexities with earlier bounds.
Table 1. Bounds on the total variation mixing time for initial laws with R 2 ( ϱ 0 ∣ μ ) ≤ R R_2(\varrho_0\mid\mu)\leq R R 2 ( ϱ 0 ∣ μ ) ≤ R R ≍ d R\asymp d R ≍ d (2) and ∇ 2 U ⪰ 0 \nabla^2U\succeq0 ∇ 2 U ⪰ 0 U U U L 2 L^2 L 2 [11 , 20 ] for ULD and of [29 Theorem 1] for BPS and ZZ, as explained after the table. The second block follows from Corollary 1.2, which also gives the full dependence on R R R α < 0 \alpha<0 α < 0 C L S I − 1 β d C_{\mathrm{LSI}}^{-1}\sqrt{\beta d} C LSI − 1 β d C L S I − 1 / 2 d C_{\mathrm{LSI}}^{-1/2}\sqrt{d} C LSI − 1/2 d C L S I ≤ β C_{\mathrm{LSI}}\leq\beta C LSI ≤ β [24 Corollary 5.5] and [26 Theorem 1.1] ; in its BPS row, the initial density relative to μ \mu μ e R e^R e R R 2 ( ϱ 0 ∣ μ ) ≤ R R_2(\varrho_0\mid\mu)\leq R R 2 ( ϱ 0 ∣ μ ) ≤ R Corollary B.3 , some Gaussian target satisfying the assumptions of the last row, with an initial law satisfying R 2 ( ϱ 0 ∣ μ ) ≤ R R_2(\varrho_0\mid\mu)\leq R R 2 ( ϱ 0 ∣ μ ) ≤ R d ≥ 2 d\geq2 d ≥ 2
As R → ∞ R\to\infty R → ∞ C L S I − 1 / 2 R log R C_{\mathrm{LSI}}^{-1/2}\sqrt{R\log R} C LSI − 1/2 R log R d \sqrt{d} d L 2 L^2 L 2 χ 2 ( ϱ t ∣ μ ) ≤ C e − λ t χ 2 ( ϱ 0 ∣ μ ) \chi^2(\varrho_t\mid\mu)\le Ce^{-\lambda t}\chi^2(\varrho_0\mid\mu) χ 2 ( ϱ t ∣ μ ) ≤ C e − λ t χ 2 ( ϱ 0 ∣ μ )
R 2 ( ϱ t ∣ μ ) ≤ log [ 1 + C e − λ t ( e R 2 ( ϱ 0 ∣ μ ) − 1 ) ] . R_2(\varrho_t\mid\mu)\le\log\left[1+Ce^{-\lambda t}\left(e^{R_2(\varrho_0\mid\mu)}-1\right)\right]. R 2 ( ϱ t ∣ μ ) ≤ log [ 1 + C e − λ t ( e R 2 ( ϱ 0 ∣ μ ) − 1 ) ] . which guarantees a fixed accuracy only after a time of order ( R + log C ) / λ (R+\log C)/\lambda ( R + log C ) / λ λ \lambda λ R R R Table 1 records this time, at R ≍ d R\asymp d R ≍ d L 2 L^2 L 2 [11 , 20 , 29 ] , which are of order C L S I \sqrt{C_{\mathrm{LSI}}} C LSI C L S I / d \sqrt{C_{\mathrm{LSI}}/d} C LSI / d C L S I / β C_{\mathrm{LSI}}/\sqrt{\beta} C LSI / β C L S I C_{\mathrm{LSI}} C LSI L 2 L^2 L 2 [29 ] are stated under a compactness condition on μ x \mu_x μ x Theorem 1.1 , with γ \gamma γ R R R C L S I − 1 / 2 R log R C_{\mathrm{LSI}}^{-1/2}\sqrt{R\log R} C LSI − 1/2 R log R log R \sqrt{\log R} log R
Following Fan, Li, and Lu [20 ] , we add to the quadratic energy a small multiple of a shifted corrector, whose time derivative dissipates the spatial component of the density through a macroscopic dissipation. A Poincaré inequality bounds this dissipation below by a multiple of the squared norm, which gives exponential χ 2 \chi^2 χ 2 R R R C L S I ( 1 + R ) C_{\mathrm{LSI}}(1+R) C LSI ( 1 + R )
bound by only a constant factor compared with zero shift. This larger shift admits a larger corrector weight, which raises the rate from order C L S I / R \sqrt{C_{\mathrm{LSI}}}/R C LSI / R C L S I / R \sqrt{C_{\mathrm{LSI}}/R} C LSI / R
Theorem 1.3 (Sharpness for the Rényi-2 divergence). Let d = 1 d=1 d = 1 α > 0 \alpha>0 α > 0 U ( x ) = α x 2 / 2 U(x)=\alpha x^2/2 U ( x ) = α x 2 /2 R \mathbb{R} R μ = N ( 0 , α − 1 ) ⊗ N ( 0 , 1 ) \mu=N(0,\alpha^{-1})\otimes N(0,1) μ = N ( 0 , α − 1 ) ⊗ N ( 0 , 1 ) R ≥ e 8 R\ge e^8 R ≥ e 8 ϱ 0 \varrho_0 ϱ 0 R 2 ( ϱ 0 ∣ μ ) = R R_2(\varrho_0\mid\mu)=R R 2 ( ϱ 0 ∣ μ ) = R γ > 0 \gamma>0 γ > 0
R 2 ( ϱ t ∣ μ ) > 1 2 for all 0 ≤ t < 1 16 α R log R . (14) R_2(\varrho_t\mid\mu)>\frac{1}{2}\qquad\text{for all }0\le t<\frac{1}{16\sqrt{\alpha}}\sqrt{R\log R}. \tag*{(14)} R 2 ( ϱ t ∣ μ ) > 2 1 for all 0 ≤ t < 16 α 1 R log R . (14) With C L S I = α C_{\mathrm{LSI}}=\alpha C LSI = α d = 1 d=1 d = 1 η = 1 / 2 \eta=1/2 η = 1/2 R 2 ( ϱ t ∣ μ ) ≤ 1 / 2 R_2(\varrho_t\mid\mu)\le1/2 R 2 ( ϱ t ∣ μ ) ≤ 1/2 α − 1 / 2 R log R \alpha^{-1/2}\sqrt{R\log R} α − 1/2 R log R Theorem 1.3 also limits the decay rate. If R 2 ( ϱ t ∣ μ ) ≤ 3 e − Λ t R R_2(\varrho_t\mid\mu)\le3e^{-\Lambda t}R R 2 ( ϱ t ∣ μ ) ≤ 3 e − Λ t R t ≥ 0 t\ge0 t ≥ 0 (8) , then evaluating at t = log ( 6 R ) / Λ t=\log(6R)/\Lambda t = log ( 6 R ) /Λ
Λ ≤ 16 α log ( 6 R ) R log R . \Lambda\le\frac{16\sqrt{\alpha}\log(6R)}{\sqrt{R\log R}}. Λ ≤ R log R 16 α log ( 6 R ) . For this target, optimizing γ \gamma γ (10) and (11) gives rates of order α / R \sqrt{\alpha/R} α / R Theorem 1.1 by more than a factor of order log R \sqrt{\log R} log R ∣ v ∣ |v| ∣ v ∣ § 5 .
For ZZ in dimension d ≥ 2 d\ge2 d ≥ 2 1 R 2 ( ϱ 0 ∣ μ ) ≤ R R_2(\varrho_0\mid\mu)\le R R 2 ( ϱ 0 ∣ μ ) ≤ R
In the first, Theorem B.1 , the target is isotropic and the initial law is displaced by R / C L S I \sqrt{R/C_{\mathrm{LSI}}} R / C LSI 1 / 4 1/4 1/4 R / C L S I \sqrt{R/C_{\mathrm{LSI}}} R / C LSI
The second result, Theorem B.2 , addresses the dependence on β \beta β U U U C L S I = 1 C_{\mathrm{LSI}}=1 C LSI = 1 x 1 − x 2 x_1-x_2 x 1 − x 2 U U U β \beta β { x 1 = x 2 } \{x_1=x_2\} { x 1 = x 2 } β \beta β β \beta β β / log ( e β ) \sqrt{\beta}/\log(e\beta) β / log ( e β ) C L S I − 1 β C_{\mathrm{LSI}}^{-1}\sqrt{\beta} C LSI − 1 β (13) up to the logarithm. Corollary B.3 combines the two examples into a lower bound on the ZZ mixing time, which shows that the ZZ entry of Corollary 1.2 in Table 1 cannot be improved by more than logarithmic factors. Since this lower bound is unbounded in β \beta β β \beta β (10) gives a BPS bound uniform in the confinement. Heuristically, a bounce reflects the velocity across ∇ U \nabla U ∇ U ∇ U \nabla U ∇ U
Related works
ULD underlies several discretized sampling algorithms; see, for example, [2 , 12 ] and the book [13 ] . The event-driven bounce mechanism of BPS originated with Peters and de With [31 ] and was developed into the BPS framework by Bouchard-Côté, Vollmer, and Doucet [10 ] . A one-dimensional Zigzag process arose as a scaling limit of lifted Metropolis–Hastings chains for the Curie–Weiss model [8 ] , and Bierkens, Fearnhead, and Roberts [6 ] developed the Zigzag process as a general-purpose sampler. Both belong to the broader class of PDMP Monte Carlo methods surveyed in [21 ] . Harris-type arguments establish geometric ergodicity of BPS and ZZ under tail and curvature conditions [9 , 14 , 17 ] . Bierkens, Kamatani, and Roberts [7 ] study the contrasting behavior of BPS and ZZ on anisotropic Gaussian targets through scaling limits.
In kinetic samplers, the position follows a velocity that is randomized only gradually, so on short time scales it moves ballistically rather than diffusively. This persistent motion can accelerate convergence. For convex potentials, ULD with optimized friction converges in L 2 L^2 L 2 C P \sqrt{C_{\mathrm{P}}} C P C P C_{\mathrm{P}} C P μ x \mu_x μ x C P C_{\mathrm{P}} C P [11 , 20 ] . Hypocoercivity explains how Hamiltonian transport turns velocity dissipation into convergence in position. Villani [32 ] developed a general theory of hypocoercivity for kinetic equations. Dolbeault, Mouhot, and Schmeiser [15 , 16 ] introduced the modified L 2 L^2 L 2 [22 ] gave a rigorous operator treatment for Langevin dynamics, and Andrieu, Durmus, Nüsken, and Roussel [3 ] developed modified L 2 L^2 L 2 [20 ] developed the shifted corrector under a Poincaré inequality of μ x \mu_x μ x
A complementary route to L 2 L^2 L 2 [11 ] obtained explicit L 2 L^2 L 2 [29 ] obtained such rates for randomized Hamiltonian Monte Carlo, BPS, and ZZ. From the lifting viewpoint, Eberle and Lörler [19 ] interpret BPS and ZZ as nonreversible lifts and show that any second-order lift can improve the relaxation time by at most a square-root factor. All these results give quantitative L 2 L^2 L 2 L 2 L^2 L 2 μ x \mu_x μ x
For ULD, Lu [25 ] established sharp hypocoercive entropy decay using the Wasserstein entropy current as a corrector. Li and Lu [24 ] developed a space-time LSI and hypocoercive hypercontractivity. For convex potentials, under an LSI of μ x \mu_x μ x q > 1 q>1 q > 1 R log R \sqrt{R\log R} R log R Theorem 1.3 rules it out for the time at which the Rényi-2 divergence falls below 1 / 2 1/2 1/2 Theorem B.1 rules it out for the total variation mixing time of ZZ in every dimension and of BPS in dimension one.
For PDMPs, Monmarché and Wang [30, Theorem 2] show that BPS and ZZ admit no uniform multiplicative contraction of the KL divergence over the initial laws whose KL divergence lies below a fixed positive level, even for a standard Gaussian target. Their theorem covers standard Gaussian velocities. Our estimates require a finite bound R R R L 2 L^2 L 2 R R R R 2 R_2 R 2
In a companion paper [26 ] , we study relative entropy decay for BPS with strongly convex and smooth potentials, using the Wasserstein entropy current of Lu [25 ] as a corrector. For initial laws with density bounded by M M M μ \mu μ M M M R 2 ( ϱ 0 ∣ μ ) ≤ log M R_2(\varrho_0 \mid\mu) \le\log M R 2 ( ϱ 0 ∣ μ ) ≤ log M
Where both results apply, the companion bound has a better dependence on the dimension and the condition number; see the BPS row of the entropy hypocoercivity block in Table 1 . The dimension enters its rate additively, alongside log M \log M log M (10) , by contrast, the dimension multiplies the divergence term or enters through β d \beta d β d R = log M R=\log M R = log M log R \sqrt{\log R} log R L 2 L^2 L 2 Proposition 3.4 . It therefore covers ULD, ZZ, and nonconvex potentials satisfying an LSI, and it should apply to other dynamics for which the modified L 2 L^2 L 2
The size of the initial divergence is a recurring issue in the complexity of sampling. Warm-start analyses of the Metropolis-adjusted Langevin algorithm assume a bound on it [18 , 33 ] , and Altschuler and Chewi [1 ] construct Rényi warm starts algorithmically using discretized ULD. For ZZ, Lu and Wang [28 ] bound the query complexity under a warm-start assumption. Our bounds concern mixing in continuous time. Lu and Luo [27 ] simulate BPS and ZZ exactly by windowed thinning, with an expected number of gradient queries proportional to the time horizon. Combining their analysis with Corollary 1.2, for 0 < α ≤ β < ∞ 0 < \alpha\le\beta< \infty 0 < α ≤ β < ∞ κ = β / α \kappa=\beta/\alpha κ = β / α δ \delta δ O ~ ( min { κ 1 / 2 d 3 / 2 , κ d } ) \widetilde{O}(\min\{\kappa^{1/2}d^{3/2},\kappa d\}) O ( min { κ 1/2 d 3/2 , κ d }) O ~ ( κ 1 / 2 d 3 / 4 + κ d 1 / 4 ) \widetilde{O}(\kappa^{1/2}d^{3/4}+\kappa d^{1/4}) O ( κ 1/2 d 3/4 + κ d 1/4 ) d d d O ~ \widetilde{O} O d d d κ \kappa κ 1 / δ 1/\delta 1/ δ O ~ ( κ 1 / 2 d 2 ) \widetilde{O}(\kappa^{1/2}d^2) O ( κ 1/2 d 2 ) O ~ ( κ d 5 / 4 ) \widetilde{O}(\kappa d^{5/4}) O ( κ d 5/4 ) [27 ] . For BPS under the same assumptions, the entropy estimate of the companion paper [26 ] gives the smaller bound O ~ ( κ 1 / 2 d ) \widetilde{O}(\kappa^{1/2}d) O ( κ 1/2 d )
Organization
Organization. Section 2 sets up the relative densities and recalls the shifted corrector and the modified energy. Section 3 establishes the dissipation of the modified energy and the nonlinear spatial resolvent estimate, and ends with the sampler-specific calculations. Section 4 proves Theorem 1.1 and Corollary 1.2. Appendix A proves Theorem 1.3, and Appendix B proves the lower bounds that show the ZZ bound to be sharp up to logarithmic factors.
AI Usage
AI Usage. During our project to improve the cold-start query complexity estimate of [27 ] , which led to the entropy estimate in [26 ] , our interaction with ChatGPT 6 Astra led to the observation that a Rényi-2 contraction estimate could improve the query complexity estimate of BPS. After checking the AI output, whose argument was rather opaque, the authors realized the underlying mechanism: the L 2 L^2 L 2 L 2 L^2 L 2
Preliminaries
Unless indicated otherwise, norms and inner products are in L 2 ( μ ) L^2(\mu) L 2 ( μ ) x x x L 2 ( μ x ) L^2(\mu_x) L 2 ( μ x ) L 2 ( μ x ) L^2(\mu_x) L 2 ( μ x )
L 0 2 ( μ ) : = { f ∈ L 2 ( μ ) : ∫ f d μ = 0 } . (15) L^2_0(\mu) := \left\{f \in L^2(\mu) : \int f\,d\mu= 0\right\}. \tag*{(15)} L 0 2 ( μ ) := { f ∈ L 2 ( μ ) : ∫ f d μ = 0 } . (15) We use the same convention for L 0 2 ( μ x ) L^2_0(\mu_x) L 0 2 ( μ x )
We identify Π v h \Pi_vh Π v h Π v \Pi_v Π v x x x L 2 ( μ ) L^2(\mu) L 2 ( μ ) C c ∞ ( R 2 d ) C^\infty_c(\mathbb{R}^{2d}) C c ∞ ( R 2 d ) C c ∞ ( R d ) C^\infty_c(\mathbb{R}^d) C c ∞ ( R d ) x x x x x x
L Π v = ( v ⋅ ∇ x ) Π v . \mathcal{L}\Pi_v = (v\cdot\nabla_x)\Pi_v. L Π v = ( v ⋅ ∇ x ) Π v . The overdamped generator L o = Δ x − ∇ U ⋅ ∇ x \mathcal{L}_o=\Delta_x-\nabla U\cdot\nabla_x L o = Δ x − ∇ U ⋅ ∇ x x x x − L o -\mathcal{L}_o − L o h ↦ ∥ ∇ x h ∥ 2 h\mapsto\|\nabla_xh\|^2 h ↦ ∥ ∇ x h ∥ 2
Relative densities
2.1. Relative densities. For the law ρ t \rho_t ρ t t t t
f t = d ρ t d μ − 1. (16) f_t=\frac{d\rho_t}{d\mu}-1. \tag*{(16)} f t = d μ d ρ t − 1. (16) We decompose it into its spatial component and velocity fluctuation:
f S , t = Π v f t , f F , t = ( I − Π v ) f t . (17) f_{S,t}=\Pi_vf_t,\qquad f_{F,t}=(I-\Pi_v)f_t. \tag*{(17)} f S , t = Π v f t , f F , t = ( I − Π v ) f t . (17) Thus f t = f S , t + f F , t f_t=f_{S,t}+f_{F,t} f t = f S , t + f F , t Π v f F , t = 0 \Pi_v f_{F,t}=0 Π v f F , t = 0 1 + f t 1+f_t 1 + f t 1 + f S , t 1+f_{S,t} 1 + f S , t ϱ t \varrho_t ϱ t μ \mu μ μ x \mu_x μ x
f t ≥ − 1 , f S , t ≥ − 1 , ∫ f t d μ = ∫ f S , t d μ x = 0. f_t\geq-1,\qquad f_{S,t}\geq-1,\qquad\int f_t\,\mathrm{d}\mu=\int f_{S,t}\,\mathrm{d}\mu_x=0. f t ≥ − 1 , f S , t ≥ − 1 , ∫ f t d μ = ∫ f S , t d μ x = 0. When no confusion arises, we suppress the time subscript and write f f f f S f_S f S f F f_F f F h h h μ \mu μ ∫ L h d μ = 0 \int\mathcal{L}h\,\mathrm{d}\mu=0 ∫ L h d μ = 0
d d t ⟨ f t , h ⟩ = ∫ L h d ϱ t = ⟨ 1 + f t , L h ⟩ = ⟨ f t , L h ⟩ . (18) \frac{\mathrm{d}}{\mathrm{d}t}\langle f_t,h\rangle=\int\mathcal{L}h\,\mathrm{d}\varrho_t=\langle1+f_t,\mathcal{L}h\rangle=\langle f_t,\mathcal{L}h\rangle. \tag*{(18)} d t d ⟨ f t , h ⟩ = ∫ L h d ϱ t = ⟨ 1 + f t , L h ⟩ = ⟨ f t , L h ⟩ . (18) Consequently,
∂ t f t = L ∗ f t . (19) \partial_t f_t=\mathcal{L}^*f_t. \tag*{(19)} ∂ t f t = L ∗ f t . (19) This equation is understood in the weak sense; if f 0 ∈ D ( L ∗ ) f_0\in D(\mathcal{L}^*) f 0 ∈ D ( L ∗ ) L 2 ( μ ) L^2(\mu) L 2 ( μ )
∥ 1 + f t ∥ 2 = 1 + ∥ f t ∥ 2 , ∥ 1 + f S , t ∥ 2 = 1 + ∥ f S , t ∥ 2 , ∥ f t ∥ 2 = ∥ f S , t ∥ 2 + ∥ f F , t ∥ 2 , R 2 ( ϱ t ∣ μ ) = log ( 1 + ∥ f t ∥ 2 ) . (20) \begin{aligned}
\lVert1+f_t\rVert^2&=1+\lVert f_t\rVert^2,\qquad&\lVert1+f_{S,t}\rVert^2&=1+\lVert f_{S,t}\rVert^2,\\
\lVert f_t\rVert^2&=\lVert f_{S,t}\rVert^2+\lVert f_{F,t}\rVert^2,\qquad&R_2(\varrho_t\mid\mu)&=\log(1+\lVert f_t\rVert^2). \tag*{(20)}
\end{aligned} ∥ 1 + f t ∥ 2 ∥ f t ∥ 2 = 1 + ∥ f t ∥ 2 , = ∥ f S , t ∥ 2 + ∥ f F , t ∥ 2 , ∥ 1 + f S , t ∥ 2 R 2 ( ϱ t ∣ μ ) = 1 + ∥ f S , t ∥ 2 , = log ( 1 + ∥ f t ∥ 2 ) . (20) The shifted corrector and modified energy
The quadratic energy 1 2 ∥ f ∥ 2 \frac{1}{2}\lVert f\rVert^2 2 1 ∥ f ∥ 2 f F f_F f F f S f_S f S L 2 L^2 L 2 [15 , 16 ] , which builds on ideas of Hérau [23 ] , quantifies this mechanism. It adds to 1 2 ∥ f ∥ 2 \frac{1}{2}\lVert f\rVert^2 2 1 ∥ f ∥ 2 f S f_S f S [20 Section 2] .
Definition 2.1 (Shifted corrector and modified energy). Let a > 0 a>0 a > 0 ϵ > 0 \epsilon>0 ϵ > 0
A a = ( a − L 0 ) − 1 ( v ⋅ ∇ x ) Π v ) ∗ . (21) \mathcal{A}_a=(a-\mathcal{L}_0)^{-1}(v\cdot\nabla_x)\Pi_v)^*. \tag*{(21)} A a = ( a − L 0 ) − 1 ( v ⋅ ∇ x ) Π v ) ∗ . (21) The corrector is defined first on test functions and extended by continuity to L 0 2 ( μ ) L^2_0(\mu) L 0 2 ( μ ) Lemma 2.2 . For f ∈ L 0 2 ( μ ) f\in L^2_0(\mu) f ∈ L 0 2 ( μ )
L a , ϵ ( f ) : = 1 2 ∥ f ∥ 2 + ϵ ⟨ A a f , f ⟩ . (22) \mathcal{L}_{a,\epsilon}(f):=\frac{1}{2}\lVert f\rVert^2+\epsilon\langle\mathcal{A}_a f,f\rangle. \tag*{(22)} L a , ϵ ( f ) := 2 1 ∥ f ∥ 2 + ϵ ⟨ A a f , f ⟩ . (22) For the law ϱ t \varrho_t ϱ t f t = d ϱ t / d μ − 1 f_t=\mathrm{d}\varrho_t/\mathrm{d}\mu-1 f t = d ϱ t / d μ − 1 (16) .
The correction enters (22) with the sign opposite to that in [20 ] . They differentiate the energy along the backward equation ∂ t f = L f \partial_t f=\mathcal{L}f ∂ t f = L f (19) , in which the transport term of the generator changes sign.
The shift a a a ϵ \epsilon ϵ α , β \alpha,\beta α , β C L S I C_{\mathrm{LSI}} C LSI γ \gamma γ Proposition 3.1 , and the shift in the proofs of § 4 .
We record operator bounds for the corrector. They also justify the differentiation of the correction in the proof of Proposition 3.1 .
Lemma 2.2 (Corrector identities and operator bounds). *For every a > 0 a>0 a > 0 Definition 2.1 has a bounded extension to L 0 2 ( μ ) L_0^2(\mu) L 0 2 ( μ )
Π v A a = A a , A a Π v = 0 , \Pi_v\mathcal{A}_a=\mathcal{A}_a,\qquad\mathcal{A}_a\Pi_v=0, Π v A a = A a , A a Π v = 0 , ∥ A a ∥ ≤ 1 2 a , ∥ ( v ⋅ ∇ x ) A a ∥ ≤ 1. (23) \|\mathcal{A}_a\|\le\frac{1}{2\sqrt{a}},\qquad\|(v\cdot\nabla_x)\mathcal{A}_a\|\le1. \tag*{(23)} ∥ A a ∥ ≤ 2 a 1 , ∥ ( v ⋅ ∇ x ) A a ∥ ≤ 1. (23) Here the norms in (23) are operator norms on L 0 2 ( μ ) L_0^2(\mu) L 0 2 ( μ ) f ∈ L 0 2 ( μ ) f\in L_0^2(\mu) f ∈ L 0 2 ( μ )
A a ∗ f = ( v ⋅ ∇ x ) ( a − L 0 ) − 1 Π v f . (24) \mathcal{A}_a^*f=(v\cdot\nabla_x)(a-\mathcal{L}_0)^{-1}\Pi_v f. \tag*{(24)} A a ∗ f = ( v ⋅ ∇ x ) ( a − L 0 ) − 1 Π v f . (24) The identities and bounds are proved in Fan, Li, and Lu [20 Section 3 and Lemma 1] . For a sufficiently small correction weight, the modified energy is comparable to the squared L 2 L^2 L 2
Lemma 2.3 (Energy equivalence). *For every a > 0 a>0 a > 0 0 < ϵ ≤ a 0<\epsilon\le\sqrt{a} 0 < ϵ ≤ a f ∈ L 0 2 ( μ ) f\in L_0^2(\mu) f ∈ L 0 2 ( μ )
1 4 ∥ f ∥ 2 ≤ L a , ϵ ( f ) ≤ 3 4 ∥ f ∥ 2 . (25) \frac{1}{4}\|f\|^2\le L_{a,\epsilon}(f)\le\frac{3}{4}\|f\|^2. \tag*{(25)} 4 1 ∥ f ∥ 2 ≤ L a , ϵ ( f ) ≤ 4 3 ∥ f ∥ 2 . (25) Proof. Since Π v f F = 0 \Pi_v f_F=0 Π v f F = 0 f S f_S f S
⟨ f F , f S ⟩ = ⟨ Π v f F , f S ⟩ = 0 , ∥ f ∥ 2 = ∥ f S ∥ 2 + ∥ f F ∥ 2 . \langle f_F,f_S\rangle=\langle\Pi_v f_F,f_S\rangle=0,\qquad\|f\|^2=\|f_S\|^2+\|f_F\|^2. ⟨ f F , f S ⟩ = ⟨ Π v f F , f S ⟩ = 0 , ∥ f ∥ 2 = ∥ f S ∥ 2 + ∥ f F ∥ 2 . Also A a f = A a f F \mathcal{A}_a f=\mathcal{A}_a f_F A a f = A a f F
∣ ϵ ⟨ A a f , f ⟩ ∣ = ∣ ϵ ⟨ A a f F , f S ⟩ ∣ ≤ ϵ 2 a ∥ f F ∥ ∥ f S ∥ ≤ ϵ 4 a ( ∥ f F ∥ 2 + ∥ f S ∥ 2 ) ≤ 1 4 ∥ f ∥ 2 . |\epsilon\langle\mathcal{A}_a f,f\rangle|=|\epsilon\langle\mathcal{A}_a f_F,f_S\rangle|
\le\frac{\epsilon}{2\sqrt{a}}\|f_F\|\|f_S\|
\le\frac{\epsilon}{4\sqrt{a}}(\|f_F\|^2+\|f_S\|^2)\le\frac{1}{4}\|f\|^2. ∣ ϵ ⟨ A a f , f ⟩ ∣ = ∣ ϵ ⟨ A a f F , f S ⟩ ∣ ≤ 2 a ϵ ∥ f F ∥∥ f S ∥ ≤ 4 a ϵ ( ∥ f F ∥ 2 + ∥ f S ∥ 2 ) ≤ 4 1 ∥ f ∥ 2 . The last inequality uses ϵ ≤ a \epsilon\le\sqrt{a} ϵ ≤ a (25) follows.
Dissipation estimates
We establish the two estimates that yield decay of the Rényi-2 divergence. The first bounds the time derivative of the modified energy by a macroscopic dissipation and the velocity dissipation. The second bounds the macroscopic dissipation below through the LSI; it is the only place where the LSI enters.
For f S ∈ L 0 2 ( μ x ) f_S\in L_0^2(\mu_x) f S ∈ L 0 2 ( μ x )
D a ( f S ) = ⟨ f S , ( − L 0 ) ( a − L 0 ) − 1 f S ⟩ . (26) \mathcal{D}_a(f_S)=\langle f_S,(-\mathcal{L}_0)(a-\mathcal{L}_0)^{-1}f_S\rangle. \tag*{(26)} D a ( f S ) = ⟨ f S , ( − L 0 ) ( a − L 0 ) − 1 f S ⟩ . (26) The name follows the macroscopic coercivity condition of [15 Assumption (H2)] and the terminology of [20 ] .
Proposition 3.1 (Dissipation of the modified energy). *Assume (1) , with β < ∞ \beta<\infty β < ∞ Z Z ZZ Z Z a > 0 a>0 a > 0 ϵ \epsilon ϵ ϱ t \varrho_t ϱ t
time t t t f t = d ϱ t / d μ − 1 f_t = \mathrm{d}\varrho_t/\mathrm{d}\mu- 1 f t = d ϱ t / d μ − 1 (16) . If f 0 ∈ D ( L ∗ ) f_0 \in D(\mathcal{L}^*) f 0 ∈ D ( L ∗ ) t ≥ 0 t \ge0 t ≥ 0
d d t L a , ϵ ( f t ) ≤ − ϵ 2 D a ( f S ) − γ 2 ∥ f F ∥ 2 , (27) \frac{\mathrm{d}}{\mathrm{d}t}\mathcal{L}_{a,\epsilon}(f_t) \leq-\frac{\epsilon}{2}\mathcal{D}_a(f_S)-\frac{\gamma}{2}\lVert f_F\rVert^2, \tag*{(27)} d t d L a , ϵ ( f t ) ≤ − 2 ϵ D a ( f S ) − 2 γ ∥ f F ∥ 2 , (27) where
ϵ = { γ a 3 a + ( − α ) + + γ 2 , U L D , γ a 100 [ γ 2 + min { d ( a + ( − α ) + ) , a + ( − α ) + + β d } ] , B P S , γ a 100 [ a + β + ( − α ) + + γ 2 ] , Z Z . (28) \epsilon=
\begin{cases}
\frac{\gamma a}{3a+(-\alpha)_+ + \gamma^2}, & \mathrm{ULD},\\[6pt]
\frac{\gamma a}{100[\gamma^2+\min\{d(a+(-\alpha)_+),a+(-\alpha)_+ + \beta d\}]}, & \mathrm{BPS},\\[6pt]
\frac{\gamma a}{100[a+\beta+(-\alpha)_+ + \gamma^2]}, & \mathrm{ZZ}.
\end{cases} \tag*{(28)} ϵ = ⎩ ⎨ ⎧ 3 a + ( − α ) + + γ 2 γ a , 100 [ γ 2 + m i n { d ( a + ( − α ) + ) , a + ( − α ) + + β d }] γ a , 100 [ a + β + ( − α ) + + γ 2 ] γ a , ULD , BPS , ZZ . (28) Moreover, the weight (28) satisfies 0 < ϵ ≤ min { γ , a / 2 } 0 < \epsilon\leq\min\{\gamma,\sqrt{a/2}\} 0 < ϵ ≤ min { γ , a /2 }
Subsection 3.1 proves Proposition 3.1 from the velocity dissipation and sampler-dependent cross-term estimates. Subsection 3.2 bounds D a \mathcal{D}_a D a
Velocity dissipation and cross terms
The following velocity dissipation estimates are taken from Fan, Li, and Lu [20 (25), (30)] for ULD and Lu and Wang [29 (30)–(32)] for BPS and ZZ.
Lemma 3.2 (Velocity dissipation). Let f ∈ L 0 2 ( μ ) f \in L^2_0(\mu) f ∈ L 0 2 ( μ )
⟨ f , L f ⟩ = − γ ∥ ∇ v f ∥ 2 ≤ − γ ∥ f F ∥ 2 , (29) \langle f,\mathcal{L}f\rangle= -\gamma\lVert\nabla_v f\rVert^2 \leq-\gamma\lVert f_F\rVert^2, \tag*{(29)} ⟨ f , L f ⟩ = − γ ∥ ∇ v f ∥ 2 ≤ − γ ∥ f F ∥ 2 , (29) where the inequality is the Gaussian Poincaré inequality in v v v
For BPS,
⟨ f , L f ⟩ = − 1 2 ∫ ( v ⋅ ∇ U ) + [ f ( x , R x v ) − f ( x , v ) ] 2 d μ − γ ∥ f F ∥ 2 \langle f,\mathcal{L}f\rangle= -\frac{1}{2}\int(v\cdot\nabla U)_+[f(x,R_xv)-f(x,v)]^2\,\mathrm{d}\mu-\gamma\lVert f_F\rVert^2 ⟨ f , L f ⟩ = − 2 1 ∫ ( v ⋅ ∇ U ) + [ f ( x , R x v ) − f ( x , v ) ] 2 d μ − γ ∥ f F ∥ 2 ≤ − γ ∥ f F ∥ 2 . (30) \leq-\gamma\lVert f_F\rVert^2. \tag*{(30)} ≤ − γ ∥ f F ∥ 2 . (30) For ZZ,
⟨ f , L f ⟩ = − 1 2 ∑ i = 1 d ∫ ( v i ∂ x i U ) + [ f ( x , F i v ) − f ( x , v ) ] 2 d μ − γ ∥ f F ∥ 2 \langle f,\mathcal{L}f\rangle= -\frac{1}{2}\sum_{i=1}^{d}\int(v_i\partial_{x_i}U)_+[f(x,F_i v)-f(x,v)]^2\,\mathrm{d}\mu-\gamma\lVert f_F\rVert^2 ⟨ f , L f ⟩ = − 2 1 i = 1 ∑ d ∫ ( v i ∂ x i U ) + [ f ( x , F i v ) − f ( x , v ) ] 2 d μ − γ ∥ f F ∥ 2 ≤ − γ ∥ f F ∥ 2 . (31) \leq-\gamma\lVert f_F\rVert^2. \tag*{(31)} ≤ − γ ∥ f F ∥ 2 . (31) One term in the derivative of the correction in (22) is the pairing ⟨ f , L ( v ⋅ ∇ x ϕ ) ⟩ \langle f,\mathcal{L}(v\cdot\nabla_x\phi)\rangle ⟨ f , L ( v ⋅ ∇ x ϕ )⟩ ϕ = ( a − L 0 ) − 1 f S \phi=(a-\mathcal{L}_0)^{-1}f_S ϕ = ( a − L 0 ) − 1 f S x x x L ( v ⋅ ∇ x ϕ ) \mathcal{L}(v\cdot\nabla_x\phi) L ( v ⋅ ∇ x ϕ ) f S f_S f S (26) . Its fluctuation pairs with f F f_F f F ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ \langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩
Lemma 3.3 (Velocity-average identity and cross-term estimates). For ϕ ∈ C ∞ ( R d ) \phi\in C^\infty(\mathbb{R}^d) ϕ ∈ C ∞ ( R d )
Π v L ( v ⋅ ∇ x ϕ ) = L 0 ϕ . (32) \Pi_v\mathcal{L}(v\cdot\nabla_x\phi)=\mathcal{L}_0\phi. \tag*{(32)} Π v L ( v ⋅ ∇ x ϕ ) = L 0 ϕ . (32) For the remaining assertions, assume (1) , with β < ∞ \beta< \infty β < ∞ a > 0 a > 0 a > 0 f ∈ L 0 2 ( μ ) f \in L^2_0(\mu) f ∈ L 0 2 ( μ ) ϕ = ( a − L 0 ) − 1 f S \phi= (a-\mathcal{L}_0)^{-1}f_S ϕ = ( a − L 0 ) − 1 f S (32) holds for this ϕ \phi ϕ
For ULD,
∣ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ ∣ ≤ 1 + ( − α ) + + γ 2 a D a ( f S ) ∥ ∇ v f ∥ . (33) \left|\langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle\right|
\leq\sqrt{1+\frac{(-\alpha)_+ + \gamma^2}{a}}\sqrt{\mathcal{D}_a(f_S)}\|\nabla_v f\|. \tag*{(33)} ∣ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩ ∣ ≤ 1 + a ( − α ) + + γ 2 D a ( f S ) ∥ ∇ v f ∥. (33) For BPS,
∣ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ ∣ ≤ 66 d + ( 30 d + 6 ) ( − α ) + + 3 γ 2 a D a ( f S ) ∥ f F ∥ . (34) \left|\langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle\right|
\leq\sqrt{66d+\frac{(30d+6)(-\alpha)_+ + 3\gamma^2}{a}}\sqrt{\mathcal{D}_a(f_S)}\|f_F\|. \tag*{(34)} ∣ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩ ∣ ≤ 66 d + a ( 30 d + 6 ) ( − α ) + + 3 γ 2 D a ( f S ) ∥ f F ∥. (34) If in addition β < ∞ \beta< \infty β < ∞
∣ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ ∣ ≤ 66 + 66 ( − α ) + + 30 β d + 3 γ 2 a D a ( f S ) ∥ f F ∥ . (35) \left|\langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle\right|
\leq\sqrt{66+\frac{66(-\alpha)_+ + 30\beta d + 3\gamma^2}{a}}\sqrt{\mathcal{D}_a(f_S)}\|f_F\|. \tag*{(35)} ∣ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩ ∣ ≤ 66 + a 66 ( − α ) + + 30 β d + 3 γ 2 D a ( f S ) ∥ f F ∥. (35) For ZZ,
∣ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ ∣ ≤ 66 + 66 ( − α ) + + 30 β + 3 γ 2 a D a ( f S ) ∥ f F ∥ . (36) \left|\langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle\right|
\leq\sqrt{66+\frac{66(-\alpha)_+ + 30\beta+ 3\gamma^2}{a}}\sqrt{\mathcal{D}_a(f_S)}\|f_F\|. \tag*{(36)} ∣ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩ ∣ ≤ 66 + a 66 ( − α ) + + 30 β + 3 γ 2 D a ( f S ) ∥ f F ∥. (36) Lemma 3.3 is proved for each sampler in Subsections 3.3.2–3.3.4.
From now on, ϕ = ( a − L 0 ) − 1 f S \phi=(a-\mathcal{L}_0)^{-1}f_S ϕ = ( a − L 0 ) − 1 f S f S ∈ L 0 2 ( μ x ) f_S\in L^2_0(\mu_x) f S ∈ L 0 2 ( μ x ) ϕ \phi ϕ D a ( f S ) \mathcal{D}_a(f_S) D a ( f S ) f S = ( a − L 0 ) ϕ f_S=(a-\mathcal{L}_0)\phi f S = ( a − L 0 ) ϕ − L 0 ϕ -\mathcal{L}_0\phi − L 0 ϕ (26) and ⟨ ϕ , − L 0 ϕ ⟩ = ∥ ∇ x ϕ ∥ 2 \langle\phi,-\mathcal{L}_0\phi\rangle=\|\nabla_x\phi\|^2 ⟨ ϕ , − L 0 ϕ ⟩ = ∥ ∇ x ϕ ∥ 2
D a ( f S ) = a ∥ ∇ x ϕ ∥ 2 + ∥ L 0 ϕ ∥ 2 . (37) \mathcal{D}_a(f_S)=a\|\nabla_x\phi\|^2+\|\mathcal{L}_0\phi\|^2. \tag*{(37)} D a ( f S ) = a ∥ ∇ x ϕ ∥ 2 + ∥ L 0 ϕ ∥ 2 . (37) Both terms on the right are nonnegative, so
∥ L 0 ϕ ∥ 2 ≤ D a ( f S ) , ∥ ∇ x ϕ ∥ 2 ≤ 1 a D a ( f S ) . (38) \|\mathcal{L}_0\phi\|^2\leq\mathcal{D}_a(f_S),\qquad\|\nabla_x\phi\|^2\leq\frac{1}{a}\mathcal{D}_a(f_S). \tag*{(38)} ∥ L 0 ϕ ∥ 2 ≤ D a ( f S ) , ∥ ∇ x ϕ ∥ 2 ≤ a 1 D a ( f S ) . (38) By (26) , Cauchy–Schwarz, and the bound on ∥ L 0 ϕ ∥ 2 \|\mathcal{L}_0\phi\|^2 ∥ L 0 ϕ ∥ 2 (38) , D a ( f S ) = ⟨ f S , − L 0 ϕ ⟩ ≤ ∥ f S ∥ D a ( f S ) \mathcal{D}_a(f_S)=\langle f_S,-\mathcal{L}_0\phi\rangle\leq\|f_S\|\sqrt{\mathcal{D}_a(f_S)} D a ( f S ) = ⟨ f S , − L 0 ϕ ⟩ ≤ ∥ f S ∥ D a ( f S ) 0 ≤ D a ( f S ) ≤ ∥ f S ∥ 2 0\leq\mathcal{D}_a(f_S)\leq\|f_S\|^2 0 ≤ D a ( f S ) ≤ ∥ f S ∥ 2 D a \mathcal{D}_a D a L 0 2 ( μ x ) L^2_0(\mu_x) L 0 2 ( μ x )
Proof of Proposition 3.1. We first differentiate the modified energy along the forward equation (19) . The calculation follows Fan, Li, and Lu [20 (41)–(48)] , with the sign change explained in Section 2.2. The forward equation gives
1 2 d d t ∥ f ∥ 2 = ⟨ f , L ∗ f ⟩ . (39) \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|f\|^2=\langle f,\mathcal{L}^*f\rangle. \tag*{(39)} 2 1 d t d ∥ f ∥ 2 = ⟨ f , L ∗ f ⟩ . (39) For test functions, ⟨ f , L ∗ f ⟩ = ⟨ L f , f ⟩ = ⟨ f , L f ⟩ \langle f,\mathcal{L}^*f\rangle=\langle\mathcal{L}f,f\rangle=\langle f,\mathcal{L}f\rangle ⟨ f , L ∗ f ⟩ = ⟨ L f , f ⟩ = ⟨ f , L f ⟩ C ∞ ( R 2 d ) C^\infty(\mathbb{R}^{2d}) C ∞ ( R 2 d ) L ∗ \mathcal{L}^* L ∗ f ∈ D ( L ∗ ) f\in D(\mathcal{L}^*) f ∈ D ( L ∗ ) ⟨ f , L f ⟩ \langle f,\mathcal{L}f\rangle ⟨ f , L f ⟩ ⟨ f , L ∗ f ⟩ \langle f,\mathcal{L}^*f\rangle ⟨ f , L ∗ f ⟩
For the correction, the product rule gives
d d t ⟨ A a f , f ⟩ = ⟨ A a ∂ t f , f ⟩ + ⟨ A a f , ∂ t f ⟩ . (40) \frac{\mathrm{d}}{\mathrm{d}t}\langle\mathcal{A}_a f,f\rangle=\langle\mathcal{A}_a\partial_t f,f\rangle+\langle\mathcal{A}_a f,\partial_t f\rangle. \tag*{(40)} d t d ⟨ A a f , f ⟩ = ⟨ A a ∂ t f , f ⟩ + ⟨ A a f , ∂ t f ⟩ . (40) Using Π v A a = A a \Pi_v\mathcal{A}_a=\mathcal{A}_a Π v A a = A a Lemma 2.2 , and ∂ t f = L ∗ f \partial_t f=\mathcal{L}^*f ∂ t f = L ∗ f
⟨ A a ∂ t f , f ⟩ = ⟨ A a ∂ t f , f S ⟩ = ⟨ ∂ t f , A a ∗ f S ⟩ = ⟨ ∂ t f , v ⋅ ∇ x ϕ ⟩ = ⟨ f , L ( v ⋅ ∇ x ϕ ) ⟩ . \begin{aligned}
\langle\mathcal{A}_a\partial_t f,f\rangle=\langle\mathcal{A}_a\partial_t f,f_S\rangle=\langle\partial_t f,\mathcal{A}_a^*f_S\rangle\\
&=\langle\partial_t f,v\cdot\nabla_x\phi\rangle=\langle f,\mathcal{L}(v\cdot\nabla_x\phi)\rangle.
\end{aligned} ⟨ A a ∂ t f , f ⟩ = ⟨ A a ∂ t f , f S ⟩ = ⟨ ∂ t f , A a ∗ f S ⟩ = ⟨ ∂ t f , v ⋅ ∇ x ϕ ⟩ = ⟨ f , L ( v ⋅ ∇ x ϕ )⟩ . Since Π v f S = f S \Pi_v f_S=f_S Π v f S = f S Π v f F = 0 \Pi_v f_F=0 Π v f F = 0 Π v \Pi_v Π v ϕ \phi ϕ (26) of D a \mathcal{D}_a D a
⟨ f , L ( v ⋅ ∇ x ϕ ) ⟩ = ⟨ f S , Π v L ( v ⋅ ∇ x ϕ ) ⟩ + ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ = ⟨ f S , L 0 ϕ ⟩ + ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ = − D a ( f S ) + ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ . \begin{aligned}
\langle f,\mathcal{L}(v\cdot\nabla_x\phi)\rangle=\langle f_S,\Pi_v\mathcal{L}(v\cdot\nabla_x\phi)\rangle+\langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle\\
&=\langle f_S,\mathcal{L}_0\phi\rangle+\langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle\\
&=-\mathcal{D}_a(f_S)+\langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle.
\end{aligned} ⟨ f , L ( v ⋅ ∇ x ϕ )⟩ = ⟨ f S , Π v L ( v ⋅ ∇ x ϕ )⟩ + ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩ = ⟨ f S , L 0 ϕ ⟩ + ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩ = − D a ( f S ) + ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩ . For the second term in (40) , the function A a f = A a f F \mathcal{A}_a f=\mathcal{A}_a f_F A a f = A a f F x x x L A a f F = ( v ⋅ ∇ x ) A a f F \mathcal{L}\mathcal{A}_a f_F=(v\cdot\nabla_x)\mathcal{A}_a f_F L A a f F = ( v ⋅ ∇ x ) A a f F
⟨ A a f , ∂ t f ⟩ = ⟨ A a f F , L ∗ f ⟩ = ⟨ L A a f F , f ⟩ = ⟨ ( v ⋅ ∇ x ) A a f F , f ⟩ = ⟨ ( v ⋅ ∇ x ) A a f F , f F ⟩ . \begin{aligned}
\langle\mathcal{A}_a f,\partial_t f\rangle=\langle\mathcal{A}_a f_F,\mathcal{L}^*f\rangle=\langle\mathcal{L}\mathcal{A}_a f_F,f\rangle\\
&=\langle(v\cdot\nabla_x)\mathcal{A}_a f_F,f\rangle=\langle(v\cdot\nabla_x)\mathcal{A}_a f_F,f_F\rangle.
\end{aligned} ⟨ A a f , ∂ t f ⟩ = ⟨ A a f F , L ∗ f ⟩ = ⟨ L A a f F , f ⟩ = ⟨( v ⋅ ∇ x ) A a f F , f ⟩ = ⟨( v ⋅ ∇ x ) A a f F , f F ⟩ . Together, these identities give
d d t ⟨ A a f , f ⟩ = − D a ( f S ) + ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ + ⟨ f F , ( v ⋅ ∇ x ) A a f F ⟩ . (41) \frac{\mathrm{d}}{\mathrm{d}t}\langle\mathcal{A}_a f,f\rangle=-\mathcal{D}_a(f_S)+\langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle
+\langle f_F,(v\cdot\nabla_x)\mathcal{A}_a f_F\rangle. \tag*{(41)} d t d ⟨ A a f , f ⟩ = − D a ( f S ) + ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩ + ⟨ f F , ( v ⋅ ∇ x ) A a f F ⟩ . (41) Lemma 2.2 controls the final term:
∣ ⟨ f F , ( v ⋅ ∇ x ) A a f F ⟩ ∣ ≤ ∥ f F ∥ ∥ ( v ⋅ ∇ x ) A a f F ∥ ≤ ∥ f F ∥ 2 . (42) \left|\langle f_F,(v\cdot\nabla_x)\mathcal{A}_a f_F\rangle\right|\leq\|f_F\|\|(v\cdot\nabla_x)\mathcal{A}_a f_F\|\leq\|f_F\|^2. \tag*{(42)} ∣ ⟨ f F , ( v ⋅ ∇ x ) A a f F ⟩ ∣ ≤ ∥ f F ∥∥ ( v ⋅ ∇ x ) A a f F ∥ ≤ ∥ f F ∥ 2 . (42) Combining (39) , (41) , and (42) , we obtain
d d t L a , ϵ ( f ) ≤ ⟨ f , L ∗ f ⟩ − ϵ D a ( f S ) + ϵ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ + ϵ ∥ f F ∥ 2 . (43) \frac{\mathrm{d}}{\mathrm{d}t}\mathcal{L}_{a,\epsilon}(f)\leq\langle f,\mathcal{L}^*f\rangle-\epsilon\mathcal{D}_a(f_S)
+\epsilon\langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle+\epsilon\|f_F\|^2. \tag*{(43)} d t d L a , ϵ ( f ) ≤ ⟨ f , L ∗ f ⟩ − ϵ D a ( f S ) + ϵ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩ + ϵ ∥ f F ∥ 2 . (43) We now absorb the cross term. First, any finite upper Hessian bound β \beta β U U U + ∞ +\infty + ∞ (28) consequently satisfies
0 < ϵ ≤ γ a a + γ 2 ≤ min { γ , a / 2 } . 0<\epsilon\leq\frac{\gamma a}{a+\gamma^2}\leq\min\{\gamma,a/2\}. 0 < ϵ ≤ a + γ 2 γ a ≤ min { γ , a /2 } . In particular, Lemma 2.3 applies.
Write D v = ∥ ∇ v f ∥ 2 \mathcal{D}_v=\|\nabla_v f\|^2 D v = ∥ ∇ v f ∥ 2 D v = ∥ f F ∥ 2 \mathcal{D}_v=\|f_F\|^2 D v = ∥ f F ∥ 2 K K K D a ( f S ) D v \sqrt{\mathcal{D}_a(f_S)\mathcal{D}_v} D a ( f S ) D v (28) . Lemmas 3.2 and 3.3 give ⟨ f , L ∗ f ⟩ ≤ − γ D v \langle f,\mathcal{L}^{*}f\rangle\le-\gamma\mathcal{D}_{v} ⟨ f , L ∗ f ⟩ ≤ − γ D v ∥ f F ∥ 2 ≤ D v \|f_{F}\|^{2}\le\mathcal{D}_{v} ∥ f F ∥ 2 ≤ D v K D a ( f S ) D v K\sqrt{\mathcal{D}_{a}(f_{S})\mathcal{D}_{v}} K D a ( f S ) D v (43) yields
d d t L a , ϵ ( f ) ≤ − ϵ D a ( f S ) − γ D v + ϵ K D a ( f S ) D v + ϵ D v ≤ − ϵ 2 D a ( f S ) − [ γ − ϵ ( 1 + K 2 2 ) ] D v . \begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{L}_{a,\epsilon}(f)\le-\epsilon\mathcal{D}_{a}(f_{S})-\gamma\mathcal{D}_{v}+\epsilon K\sqrt{\mathcal{D}_{a}(f_{S})\mathcal{D}_{v}}+\epsilon\mathcal{D}_{v} \\
&\le-\frac{\epsilon}{2}\mathcal{D}_{a}(f_{S})-\left[\gamma-\epsilon\left(1+\frac{K^{2}}{2}\right)\right]\mathcal{D}_{v}.
\end{aligned} d t d L a , ϵ ( f ) ≤ − ϵ D a ( f S ) − γ D v + ϵK D a ( f S ) D v + ϵ D v ≤ − 2 ϵ D a ( f S ) − [ γ − ϵ ( 1 + 2 K 2 ) ] D v . It remains to check ϵ ( K 2 + 2 ) ≤ γ \epsilon(K^{2}+2)\le\gamma ϵ ( K 2 + 2 ) ≤ γ
a ( K 2 + 2 ) ≤ { 3 a + ( − α ) + + γ 2 , U L D , 100 [ γ 2 + d ( a + ( − α ) + ) ] , B P S , x r e f ‘ ( 3.9 ) ‘ , 100 [ γ 2 + a + ( − α ) + + β d ] , B P S , x r e f ‘ ( 3.10 ) ‘ , 100 [ γ 2 + a + ( − α ) + + β ] , Z Z . a(K^{2}+2)\le
\begin{cases}
3a+(-\alpha)_{+}+\gamma^{2}, & \mathrm{ULD},\\
100[\gamma^{2}+d(a+(-\alpha)_{+})], & \mathrm{BPS},\ {xref}`(3.9)`,\\
100[\gamma^{2}+a+(-\alpha)_{+}+\beta d], & \mathrm{BPS},\ {xref}`(3.10)`,\\
100[\gamma^{2}+a+(-\alpha)_{+}+\beta], & \mathrm{ZZ}.
\end{cases} a ( K 2 + 2 ) ≤ ⎩ ⎨ ⎧ 3 a + ( − α ) + + γ 2 , 100 [ γ 2 + d ( a + ( − α ) + )] , 100 [ γ 2 + a + ( − α ) + + β d ] , 100 [ γ 2 + a + ( − α ) + + β ] , ULD , BPS , x r e f ‘ ( 3.9 ) ‘ , BPS , x r e f ‘ ( 3.10 ) ‘ , ZZ . For (34) , use 66 d + 2 ≤ 68 d 66d+2\le68d 66 d + 2 ≤ 68 d 30 d + 6 ≤ 36 d 30d+6\le36d 30 d + 6 ≤ 36 d γ a / ϵ \gamma a/\epsilon γ a / ϵ (28) . Thus the remaining velocity coefficient is at least γ / 2 \gamma/2 γ /2 D v ≥ ∥ f F ∥ 2 \mathcal{D}_{v}\ge\|f_{F}\|^{2} D v ≥ ∥ f F ∥ 2 (27) . □
A nonlinear spatial resolvent estimate
We now use the LSI to bound the macroscopic dissipation D a \mathcal{D}_{a} D a (26) , from below. The Poincaré inequality implied by (2) and spectral calculus give the linear estimate
D a ( f S ) ≥ C L S I a + C L S I ∥ f S ∥ 2 . (44) \mathcal{D}_{a}(f_{S})\ge\frac{C_{\mathrm{LSI}}}{a+C_{\mathrm{LSI}}}\|f_{S}\|^{2}. \tag*{(44)} D a ( f S ) ≥ a + C LSI C LSI ∥ f S ∥ 2 . (44) Together with Proposition 3.1, it yields exponential decay of χ 2 \chi^{2} χ 2 R R R 1 + f S 1+f_{S} 1 + f S ∥ f ∥ 2 \|f\|^{2} ∥ f ∥ 2 c ( 1 + ∥ f ∥ 2 ) log ( 1 + ∥ f ∥ 2 ) c(1+\|f\|^{2})\log(1+\|f\|^{2}) c ( 1 + ∥ f ∥ 2 ) log ( 1 + ∥ f ∥ 2 ) log ( 1 + ∥ f ∥ 2 ) \log(1+\|f\|^{2}) log ( 1 + ∥ f ∥ 2 ) c c c
Proposition 3.4 (Nonlinear spatial resolvent estimate). Assume (2) . For every a > 0 a>0 a > 0 and every f S ∈ L 0 2 ( μ x ) f_{S}\in L^{2}_{0}(\mu_{x}) f S ∈ L 0 2 ( μ x ) satisfying f S ≥ − 1 f_{S}\ge-1 f S ≥ − 1
D a ( f S ) ≥ C L S I ( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) 2 a + C L S I ( 1 + log ( 1 + ∥ f S ∥ 2 ) ) . (45) \mathcal{D}_{a}(f_{S})\ge\frac{C_{\mathrm{LSI}}(1+\|f_{S}\|^{2})\log(1+\|f_{S}\|^{2})}{2a+C_{\mathrm{LSI}}(1+\log(1+\|f_{S}\|^{2}))}. \tag*{(45)} D a ( f S ) ≥ 2 a + C LSI ( 1 + log ( 1 + ∥ f S ∥ 2 )) C LSI ( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) . (45) Proof. We apply the LSI to a density obtained by smoothing 1 + f S 1+f_{S} 1 + f S a D a ( f S ) a\mathcal{D}_{a}(f_{S}) a D a ( f S )
ψ = a ( a − L o ) − 1 / 2 ( 1 + f S ) = a π ∫ 0 ∞ t − 1 / 2 e − a t e t L o ( 1 + f S ) d t . (46) \psi=\sqrt{a}(a-\mathcal{L}_{o})^{-1/2}(1+f_{S})=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}t^{-1/2}e^{-at}e^{t\mathcal{L}_{o}}(1+f_{S})\,\mathrm{d}t. \tag*{(46)} ψ = a ( a − L o ) − 1/2 ( 1 + f S ) = π a ∫ 0 ∞ t − 1/2 e − a t e t L o ( 1 + f S ) d t . (46) The second equality follows from spectral calculus, since the spectrum of a − L o a-\mathcal{L}_{o} a − L o [ a , ∞ ) [a,\infty) [ a , ∞ )
λ − 1 / 2 = 1 π ∫ 0 ∞ t − 1 / 2 e − λ t d t , λ > 0. \lambda^{-1/2}=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}t^{-1/2}e^{-\lambda t}\,\mathrm{d}t,\qquad\lambda>0. λ − 1/2 = π 1 ∫ 0 ∞ t − 1/2 e − λ t d t , λ > 0. Taking λ = a \lambda=a λ = a a / π t − 1 / 2 e − a t \sqrt{a/\pi}\,t^{-1/2}e^{-at} a / π t − 1/2 e − a t e t L o e^{t\mathcal{L}_{o}} e t L o L 2 ( μ x ) L^{2}(\mu_{x}) L 2 ( μ x ) L 2 ( μ x ) L^{2}(\mu_{x}) L 2 ( μ x ) ψ \psi ψ L 0 1 = 0 \mathcal{L}_0 1=0 L 0 1 = 0 ψ = 1 + a ( a − L 0 ) − 1 / 2 f S \psi=1+\sqrt{a}(a-\mathcal{L}_0)^{-1/2}f_S ψ = 1 + a ( a − L 0 ) − 1/2 f S f S f_S f S ( − L 0 ) 1 / 2 ( a − L 0 ) − 1 / 2 (-\mathcal{L}_0)^{1/2}(a-\mathcal{L}_0)^{-1/2} ( − L 0 ) 1/2 ( a − L 0 ) − 1/2 ψ \psi ψ h ↦ ∥ ∇ x h ∥ 2 h\mapsto\|\nabla_x h\|^2 h ↦ ∥ ∇ x h ∥ 2 (26) and a ( a − L 0 ) − 1 = I − ( − L 0 ) ( a − L 0 ) − 1 a(a-\mathcal{L}_0)^{-1}=I-(-\mathcal{L}_0)(a-\mathcal{L}_0)^{-1} a ( a − L 0 ) − 1 = I − ( − L 0 ) ( a − L 0 ) − 1
∥ ∇ x ψ ∥ 2 = a ⟨ f S , ( − L 0 ) ( a − L 0 ) − 1 f S ⟩ = a D a ( f S ) , ∥ ψ ∥ 2 = 1 + a ⟨ f S , ( a − L 0 ) − 1 f S ⟩ = 1 + ∥ f S ∥ 2 − D a ( f S ) . (47) \begin{aligned}
\|\nabla_x\psi\|^2&=a\langle f_S,(-\mathcal{L}_0)(a-\mathcal{L}_0)^{-1}f_S\rangle=a\mathcal{D}_a(f_S),\\
\|\psi\|^2&=1+a\langle f_S,(a-\mathcal{L}_0)^{-1}f_S\rangle=1+\|f_S\|^2-\mathcal{D}_a(f_S). \tag*{(47)}
\end{aligned} ∥ ∇ x ψ ∥ 2 ∥ ψ ∥ 2 = a ⟨ f S , ( − L 0 ) ( a − L 0 ) − 1 f S ⟩ = a D a ( f S ) , = 1 + a ⟨ f S , ( a − L 0 ) − 1 f S ⟩ = 1 + ∥ f S ∥ 2 − D a ( f S ) . (47) We next bound Ent μ x ( ψ 2 ) \operatorname{Ent}_{\mu_x}(\psi^2) Ent μ x ( ψ 2 ) ψ \psi ψ u ↦ u log u u\mapsto u\log u u ↦ u log u ψ d μ x \psi\,d\mu_x ψ d μ x
∫ ψ 2 log ψ 2 d μ x = 2 ∫ ψ ( ψ log ψ ) d μ x ≥ 2 ∥ ψ ∥ 2 log ∥ ψ ∥ . \int\psi^2\log\psi^2\,d\mu_x=2\int\psi(\psi\log\psi)\,d\mu_x\geq2\|\psi\|^2\log\|\psi\|. ∫ ψ 2 log ψ 2 d μ x = 2 ∫ ψ ( ψ log ψ ) d μ x ≥ 2∥ ψ ∥ 2 log ∥ ψ ∥. Subtracting ∥ ψ ∥ 2 log ∥ ψ ∥ 2 \|\psi\|^2\log\|\psi\|^2 ∥ ψ ∥ 2 log ∥ ψ ∥ 2 ∥ ∇ x ψ ∥ 2 \|\nabla_x\psi\|^2 ∥ ∇ x ψ ∥ 2 (47) :
∥ ψ ∥ 2 log ∥ ψ ∥ 2 ≤ Ent μ x ( ψ 2 ) = ∫ ψ 2 log ψ 2 d μ x − ∥ ψ ∥ 2 log ∥ ψ ∥ 2 ≤ 2 a C L S I D a ( f S ) . (48) \|\psi\|^2\log\|\psi\|^2\leq\operatorname{Ent}_{\mu_x}(\psi^2)=\int\psi^2\log\psi^2\,d\mu_x-\|\psi\|^2\log\|\psi\|^2\leq\frac{2a}{C_{\mathrm{LSI}}}\mathcal{D}_a(f_S). \tag*{(48)} ∥ ψ ∥ 2 log ∥ ψ ∥ 2 ≤ Ent μ x ( ψ 2 ) = ∫ ψ 2 log ψ 2 d μ x − ∥ ψ ∥ 2 log ∥ ψ ∥ 2 ≤ C LSI 2 a D a ( f S ) . (48) The convexity of u log u u\log u u log u ∥ ψ ∥ 2 \|\psi\|^2 ∥ ψ ∥ 2 (47) , and (48) give
( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) ≤ ∥ ψ ∥ 2 log ∥ ψ ∥ 2 + [ 1 + log ( 1 + ∥ f S ∥ 2 ) ] D a ( f S ) ≤ [ 2 a C L S I + 1 + log ( 1 + ∥ f S ∥ 2 ) ] D a ( f S ) . \begin{aligned}
(1+\|f_S\|^2)\log(1+\|f_S\|^2)\leq\|\psi\|^2\log\|\psi\|^2+[1+\log(1+\|f_S\|^2)]\mathcal{D}_a(f_S) \\
&\leq\left[\frac{2a}{C_{\mathrm{LSI}}}+1+\log(1+\|f_S\|^2)\right]\mathcal{D}_a(f_S).
\end{aligned} ( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) ≤ ∥ ψ ∥ 2 log ∥ ψ ∥ 2 + [ 1 + log ( 1 + ∥ f S ∥ 2 )] D a ( f S ) ≤ [ C LSI 2 a + 1 + log ( 1 + ∥ f S ∥ 2 ) ] D a ( f S ) . Rearranging proves (45) . □
Proofs of the sampler-specific estimates
We now prove Lemma 3.3 for each sampler. All calculations are initially on test functions and extend to the resolvent functions by standard approximation.
The Bochner identity, a weighted estimate, and Gaussian moments
The integrated Bochner identity [20 (31)] reads
∥ L 0 ϕ ∥ 2 = ∥ ∇ x 2 ϕ ∥ 2 + ∫ ∇ x 2 U [ ∇ x ϕ , ∇ x ϕ ] d μ x . (49) \|\mathcal{L}_0\phi\|^2=\|\nabla_x^2\phi\|^2+\int\nabla_x^2U[\nabla_x\phi,\nabla_x\phi]\,d\mu_x. \tag*{(49)} ∥ L 0 ϕ ∥ 2 = ∥ ∇ x 2 ϕ ∥ 2 + ∫ ∇ x 2 U [ ∇ x ϕ , ∇ x ϕ ] d μ x . (49) The lower Hessian bound therefore gives
∥ ∇ x 2 ϕ ∥ 2 ≤ ∥ L 0 ϕ ∥ 2 + ( − α ) + ∥ ∇ x ϕ ∥ 2 . (50) \|\nabla_x^2\phi\|^2\leq\|\mathcal{L}_0\phi\|^2+(-\alpha)_+\|\nabla_x\phi\|^2. \tag*{(50)} ∥ ∇ x 2 ϕ ∥ 2 ≤ ∥ L 0 ϕ ∥ 2 + ( − α ) + ∥ ∇ x ϕ ∥ 2 . (50) The additional term records the contribution of negative curvature. We retain it in each sampler-specific estimate below; it vanishes when α ≥ 0 \alpha\geq0 α ≥ 0 (38) , this estimate applies for every a > 0 a>0 a > 0 a > ( − α ) + a>(-\alpha)_+ a > ( − α ) +
When β < ∞ \beta<\infty β < ∞ u ∈ C c ∞ ( R d ) u\in C^\infty_c(\mathbb{R}^d) u ∈ C c ∞ ( R d ) i i i
∥ ( ∂ x i U ) u ∥ 2 ≤ 2 β ∥ u ∥ 2 + 4 ∥ ∂ x i u ∥ 2 . (51) \|(\partial_{x_i}U)u\|^2\leq2\beta\|u\|^2+4\|\partial_{x_i}u\|^2. \tag*{(51)} ∥ ( ∂ x i U ) u ∥ 2 ≤ 2 β ∥ u ∥ 2 + 4∥ ∂ x i u ∥ 2 . (51) Integration by parts and Young’s inequality give
∥ ( ∂ x i U ) u ∥ 2 = ∫ ( ∂ x i x i U ) u 2 d μ x + 2 ∫ ( ∂ x i U ) u ∂ x i u d μ x ≤ β ∥ u ∥ 2 + 1 2 ∥ ( ∂ x i U ) u ∥ 2 + 2 ∥ ∂ x i u ∥ 2 . \|(\partial_{x_i}U)u\|^2 = \int(\partial_{x_i x_i}U)u^2\,d\mu_x + 2\int(\partial_{x_i}U)u\partial_{x_i}u\,d\mu_x
\le\beta\|u\|^2 + \frac{1}{2}\|(\partial_{x_i}U)u\|^2 + 2\|\partial_{x_i}u\|^2. ∥ ( ∂ x i U ) u ∥ 2 = ∫ ( ∂ x i x i U ) u 2 d μ x + 2 ∫ ( ∂ x i U ) u ∂ x i u d μ x ≤ β ∥ u ∥ 2 + 2 1 ∥ ( ∂ x i U ) u ∥ 2 + 2∥ ∂ x i u ∥ 2 . Absorbing the repeated term proves (51) .
We use the elementary Gaussian moment identities
∫ v i v j d μ v = δ i j , ∫ v i v j v k v l d μ v = δ i j δ k l + δ i k δ j l + δ i l δ j k . \int v_i v_j\,d\mu_v=\delta_{ij},\qquad\int v_i v_j v_k v_l\,d\mu_v=\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}. ∫ v i v j d μ v = δ ij , ∫ v i v j v k v l d μ v = δ ij δ k l + δ ik δ j l + δ i l δ j k . Since ∇ x 2 ϕ \nabla_x^2\phi ∇ x 2 ϕ x x x
∫ ( v T ∇ x 2 ϕ v − Δ x ϕ ) 2 d μ v = ∑ i , j , k , l = 1 d ( ∂ x i x j ϕ ) ( ∂ x k x l ϕ ) ∫ ( v i v j − δ i j ) ( v k v l − δ k l ) d μ v = 2 ∑ i , j = 1 d ∣ ∂ x i x j ϕ ∣ 2 = 2 ∣ ∇ x 2 ϕ ∣ H S 2 . \int(v^\mathsf{T}\nabla_x^2\phi\,v-\Delta_x\phi)^2\,d\mu_v
= \sum_{i,j,k,l=1}^{d}(\partial_{x_i x_j}\phi)(\partial_{x_k x_l}\phi)\int(v_i v_j-\delta_{ij})(v_k v_l-\delta_{kl})\,d\mu_v
=2\sum_{i,j=1}^{d}|\partial_{x_i x_j}\phi|^2=2|\nabla_x^2\phi|_{\mathrm{HS}}^2. ∫ ( v T ∇ x 2 ϕ v − Δ x ϕ ) 2 d μ v = i , j , k , l = 1 ∑ d ( ∂ x i x j ϕ ) ( ∂ x k x l ϕ ) ∫ ( v i v j − δ ij ) ( v k v l − δ k l ) d μ v = 2 i , j = 1 ∑ d ∣ ∂ x i x j ϕ ∣ 2 = 2∣ ∇ x 2 ϕ ∣ HS 2 . Likewise, ∫ ( v ⋅ ∇ x ϕ ) 2 d μ v = ∣ ∇ x ϕ ∣ 2 \int(v\cdot\nabla_x\phi)^2\,d\mu_v=|\nabla_x\phi|^2 ∫ ( v ⋅ ∇ x ϕ ) 2 d μ v = ∣ ∇ x ϕ ∣ 2 n n n ∫ ( v ⋅ n ) + 2 d μ v = 1 / 2 \int(v\cdot n)_+^2\,d\mu_v=1/2 ∫ ( v ⋅ n ) + 2 d μ v = 1/2 ∫ ( v ⋅ n ) + 4 d μ v = 3 / 2 \int(v\cdot n)_+^4\,d\mu_v=3/2 ∫ ( v ⋅ n ) + 4 d μ v = 3/2
∫ ( 2 ( v ⋅ n ) + 2 − 1 ) 2 d μ v = 4 ⋅ 3 2 − 4 ⋅ 1 2 + 1 = 5. \int\left(2(v\cdot n)_+^2-1\right)^2\,d\mu_v=4\cdot\frac{3}{2}-4\cdot\frac{1}{2}+1=5. ∫ ( 2 ( v ⋅ n ) + 2 − 1 ) 2 d μ v = 4 ⋅ 2 3 − 4 ⋅ 2 1 + 1 = 5. Estimates for underdamped Langevin dynamics
Proof of Lemma 3.3, ULD part. Applying each part of (3) to v ⋅ ∇ x ϕ v\cdot\nabla_x\phi v ⋅ ∇ x ϕ v v v
( v ⋅ ∇ x ) ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v , (v\cdot\nabla_x)(v\cdot\nabla_x\phi)=v^\mathsf{T}\nabla_x^2\phi\,v, ( v ⋅ ∇ x ) ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v , ( − ∇ U ⋅ ∇ v ) ( v ⋅ ∇ x ϕ ) = − ∇ U ⋅ ∇ x ϕ , (-\nabla U\cdot\nabla_v)(v\cdot\nabla_x\phi)=-\nabla U\cdot\nabla_x\phi, ( − ∇ U ⋅ ∇ v ) ( v ⋅ ∇ x ϕ ) = − ∇ U ⋅ ∇ x ϕ , ( Δ v − v ⋅ ∇ v ) ( v ⋅ ∇ x ϕ ) = − v ⋅ ∇ x ϕ . (\Delta_v-v\cdot\nabla_v)(v\cdot\nabla_x\phi)=-v\cdot\nabla_x\phi. ( Δ v − v ⋅ ∇ v ) ( v ⋅ ∇ x ϕ ) = − v ⋅ ∇ x ϕ . Since ∫ v i d μ v = 0 \int v_i\,d\mu_v=0 ∫ v i d μ v = 0 ∫ v i v j d μ v = δ i j \int v_i v_j\,d\mu_v=\delta_{ij} ∫ v i v j d μ v = δ ij (32) . Subtracting this average gives
( I − Π v ) L ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v − Δ x ϕ − γ v ⋅ ∇ x ϕ . (52) (I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)=v^\mathsf{T}\nabla_x^2\phi\,v-\Delta_x\phi-\gamma v\cdot\nabla_x\phi. \tag*{(52)} ( I − Π v ) L ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v − Δ x ϕ − γ v ⋅ ∇ x ϕ . (52) In particular, the force term disappears under I − Π v I-\Pi_v I − Π v
To estimate the pairing with f F f_F f F x x x
∫ f F ( v i v j − δ i j ) d μ v = ∫ ∂ v i ( v j f F ) d μ v − δ i j ∫ f F d μ v = ∫ v j ∂ v i f d μ v , \int f_F(v_i v_j-\delta_{ij})\,d\mu_v=\int\partial_{v_i}(v_jf_F)\,d\mu_v-\delta_{ij}\int f_F\,d\mu_v=\int v_j\partial_{v_i}f\,d\mu_v, ∫ f F ( v i v j − δ ij ) d μ v = ∫ ∂ v i ( v j f F ) d μ v − δ ij ∫ f F d μ v = ∫ v j ∂ v i f d μ v , ∫ f F v i d μ v = ∫ ∂ v i f d μ v . \int f_Fv_i\,d\mu_v=\int\partial_{v_i}f\,d\mu_v. ∫ f F v i d μ v = ∫ ∂ v i f d μ v . Multiplying the first identity by ∂ x i x j ϕ \partial_{x_i x_j}\phi ∂ x i x j ϕ − γ ∂ x i ϕ -\gamma\partial_{x_i}\phi − γ ∂ x i ϕ x x x
⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ = ⟨ ∇ v f , ( ∇ x 2 ϕ ) v − γ ∇ x ϕ ⟩ . (53) \langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle=\langle\nabla_v f,(\nabla_x^2\phi)v-\gamma\nabla_x\phi\rangle. \tag*{(53)} ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩ = ⟨ ∇ v f , ( ∇ x 2 ϕ ) v − γ ∇ x ϕ ⟩ . (53) The Gaussian first and second moments also imply
∥ ( ∇ x 2 ϕ ) v − γ ∇ x ϕ ∥ 2 = ∫ ∑ i , j , k = 1 d ( ∂ x i x j ϕ ) ( ∂ x i x k ϕ ) v j v k d μ + γ 2 ∥ ∇ x ϕ ∥ 2 − 2 γ ∫ ∑ i , j = 1 d ( ∂ x i x j ϕ ) ( ∂ x i ϕ ) v j d μ = ∥ ∇ x 2 ϕ ∥ 2 + γ 2 ∥ ∇ x ϕ ∥ 2 . \begin{aligned}
\left\|(\nabla_x^2\phi)v-\gamma\nabla_x\phi\right\|^2
&=\int\sum_{i,j,k=1}^{d}(\partial_{x_i x_j}\phi)(\partial_{x_i x_k}\phi)v_jv_k\,\mathrm{d}\mu+\gamma^2\|\nabla_x\phi\|^2\\
&\quad-2\gamma\int\sum_{i,j=1}^{d}(\partial_{x_i x_j}\phi)(\partial_{x_i}\phi)v_j\,\mathrm{d}\mu\\
&=\|\nabla_x^2\phi\|^2+\gamma^2\|\nabla_x\phi\|^2.
\end{aligned} ( ∇ x 2 ϕ ) v − γ ∇ x ϕ 2 = ∫ i , j , k = 1 ∑ d ( ∂ x i x j ϕ ) ( ∂ x i x k ϕ ) v j v k d μ + γ 2 ∥ ∇ x ϕ ∥ 2 − 2 γ ∫ i , j = 1 ∑ d ( ∂ x i x j ϕ ) ( ∂ x i ϕ ) v j d μ = ∥ ∇ x 2 ϕ ∥ 2 + γ 2 ∥ ∇ x ϕ ∥ 2 . The Hessian bound (50) , the resolvent estimates (38) , and Cauchy–Schwarz in (53) give
∣ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ⟩ ∣ ≤ ∥ ∇ v f ∥ ∥ ∇ x 2 ϕ ∥ 2 + γ 2 ∥ ∇ x ϕ ∥ 2 ≤ 1 + ( − α ) + + γ 2 a D a ( f S ) ∥ ∇ v f ∥ . \begin{aligned}
\left|\langle f_F,(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rangle\right|
\leq\|\nabla_v f\|\sqrt{\|\nabla_x^2\phi\|^2+\gamma^2\|\nabla_x\phi\|^2} \\
&\leq\sqrt{1+\frac{(-\alpha)_++\gamma^2}{a}}\sqrt{\mathcal{D}_a(f_S)}\|\nabla_v f\|.
\end{aligned} ∣ ⟨ f F , ( I − Π v ) L ( v ⋅ ∇ x ϕ )⟩ ∣ ≤ ∥ ∇ v f ∥ ∥ ∇ x 2 ϕ ∥ 2 + γ 2 ∥ ∇ x ϕ ∥ 2 ≤ 1 + a ( − α ) + + γ 2 D a ( f S ) ∥ ∇ v f ∥. This proves (33) . □
Estimates for the bouncy particle sampler.
Proof of Lemma 3.3, BPS part. The reflection formula gives
( R x v ) ⋅ ∇ x ϕ − v ⋅ ∇ x ϕ = − 2 ( v ⋅ n ) ( n ⋅ ∇ x ϕ ) . (R_xv)\cdot\nabla_x\phi-v\cdot\nabla_x\phi=-2(v\cdot n)(n\cdot\nabla_x\phi). ( R x v ) ⋅ ∇ x ϕ − v ⋅ ∇ x ϕ = − 2 ( v ⋅ n ) ( n ⋅ ∇ x ϕ ) . Since ∇ U = ∣ ∇ U ∣ n \nabla U=|\nabla U|n ∇ U = ∣∇ U ∣ n
( v ⋅ ∇ U ) + [ ( R x v ) ⋅ ∇ x ϕ − v ⋅ ∇ x ϕ ] = − 2 ( v ⋅ n ) + 2 ∇ U ⋅ ∇ x ϕ . (v\cdot\nabla U)_+\left[(R_xv)\cdot\nabla_x\phi-v\cdot\nabla_x\phi\right]=-2(v\cdot n)_+^2\nabla U\cdot\nabla_x\phi. ( v ⋅ ∇ U ) + [ ( R x v ) ⋅ ∇ x ϕ − v ⋅ ∇ x ϕ ] = − 2 ( v ⋅ n ) + 2 ∇ U ⋅ ∇ x ϕ . The transport and refreshment terms are computed directly, giving
L ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v − 2 ( v ⋅ n ) + 2 ∇ U ⋅ ∇ x ϕ − γ v ⋅ ∇ x ϕ . (54) \mathcal{L}(v\cdot\nabla_x\phi)=v^\mathsf{T}\nabla_x^2\phi v-2(v\cdot n)_+^2\nabla U\cdot\nabla_x\phi-\gamma v\cdot\nabla_x\phi. \tag*{(54)} L ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v − 2 ( v ⋅ n ) + 2 ∇ U ⋅ ∇ x ϕ − γ v ⋅ ∇ x ϕ . (54) Since ∫ ( v ⋅ n ) + 2 d μ v = 1 / 2 \int(v\cdot n)_+^2\,\mathrm{d}\mu_v=1/2 ∫ ( v ⋅ n ) + 2 d μ v = 1/2 n n n § 3.3.1 ), the velocity average of (54) is Δ x ϕ − ∇ U ⋅ ∇ x ϕ \Delta_x\phi-\nabla U\cdot\nabla_x\phi Δ x ϕ − ∇ U ⋅ ∇ x ϕ (32) for BPS and yields
( I − Π v ) L ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v − Δ x ϕ − ( 2 ( v ⋅ n ) + 2 − 1 ) ∇ U ⋅ ∇ x ϕ − γ v ⋅ ∇ x ϕ . (55) (I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)=v^\mathsf{T}\nabla_x^2\phi v-\Delta_x\phi-\left(2(v\cdot n)_+^2-1\right)\nabla U\cdot\nabla_x\phi-\gamma v\cdot\nabla_x\phi. \tag*{(55)} ( I − Π v ) L ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v − Δ x ϕ − ( 2 ( v ⋅ n ) + 2 − 1 ) ∇ U ⋅ ∇ x ϕ − γ v ⋅ ∇ x ϕ . (55) The Gaussian moments in § 3.3.1 control the three terms. Using ( r + s + t ) 2 ≤ 3 ( r 2 + s 2 + t 2 ) (r+s+t)^2\leq3(r^2+s^2+t^2) ( r + s + t ) 2 ≤ 3 ( r 2 + s 2 + t 2 ) (55) and integrating proves
∥ ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ∥ 2 ≤ 6 ∥ ∇ x 2 ϕ ∥ 2 + 15 ∥ ∇ U ⋅ ∇ x ϕ ∥ 2 + 3 γ 2 ∥ ∇ x ϕ ∥ 2 . (56) \left\|(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\right\|^2\leq6\|\nabla_x^2\phi\|^2+15\|\nabla U\cdot\nabla_x\phi\|^2+3\gamma^2\|\nabla_x\phi\|^2. \tag*{(56)} ∥ ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ∥ 2 ≤ 6∥ ∇ x 2 ϕ ∥ 2 + 15∥∇ U ⋅ ∇ x ϕ ∥ 2 + 3 γ 2 ∥ ∇ x ϕ ∥ 2 . (56) We bound the force term ∥ ∇ U ⋅ ∇ x ϕ ∥ 2 \|\nabla U\cdot\nabla_x\phi\|^2 ∥∇ U ⋅ ∇ x ϕ ∥ 2 (34) and (35) . For (34) , we use only the lower Hessian bound (50) . Pointwise,
∣ Δ x ϕ ∣ 2 = ∣ ∑ i = 1 d ∂ x i x i ϕ ∣ 2 ≤ d ∑ i = 1 d ∣ ∂ x i x i ϕ ∣ 2 ≤ d ∣ ∇ x 2 ϕ ∣ H S 2 . |\Delta_x\phi|^2=\left|\sum_{i=1}^{d}\partial_{x_i x_i}\phi\right|^2\leq d\sum_{i=1}^{d}|\partial_{x_i x_i}\phi|^2\leq d|\nabla_x^2\phi|_{\mathrm{HS}}^2. ∣ Δ x ϕ ∣ 2 = i = 1 ∑ d ∂ x i x i ϕ 2 ≤ d i = 1 ∑ d ∣ ∂ x i x i ϕ ∣ 2 ≤ d ∣ ∇ x 2 ϕ ∣ HS 2 . Using ∇ U ⋅ ∇ x ϕ = Δ x ϕ − L 0 ϕ \nabla U \cdot\nabla_x\phi=\Delta_x\phi-\mathcal{L}_0\phi ∇ U ⋅ ∇ x ϕ = Δ x ϕ − L 0 ϕ
∥ ∇ U ⋅ ∇ x ϕ ∥ 2 ≤ 2 ∥ Δ x ϕ ∥ 2 + 2 ∥ L 0 ϕ ∥ 2 ≤ 2 d ∥ ∇ x 2 ϕ ∥ 2 + 2 ∥ L 0 ϕ ∥ 2 ≤ 2 ( d + 1 ) ∥ L 0 ϕ ∥ 2 + 2 d ( − α ) + + ∥ ∇ x ϕ ∥ 2 . (57) \begin{aligned}
\lVert\nabla U\cdot\nabla_x\phi\rVert^2 &\le2\lVert\Delta_x\phi\rVert^2+2\lVert\mathcal{L}_0\phi\rVert^2\\
&\le2d\lVert\nabla_x^2\phi\rVert^2+2\lVert\mathcal{L}_0\phi\rVert^2\\
&\le2(d+1)\lVert\mathcal{L}_0\phi\rVert^2+2d(-\alpha)_++\lVert\nabla_x\phi\rVert^2. \tag*{(57)}
\end{aligned} ∥ ∇ U ⋅ ∇ x ϕ ∥ 2 ≤ 2 ∥ Δ x ϕ ∥ 2 + 2 ∥ L 0 ϕ ∥ 2 ≤ 2 d ∥ ∇ x 2 ϕ ∥ 2 + 2 ∥ L 0 ϕ ∥ 2 ≤ 2 ( d + 1 ) ∥ L 0 ϕ ∥ 2 + 2 d ( − α ) + + ∥ ∇ x ϕ ∥ 2 . (57) Substituting in (56) , and then using (38) , gives
∥ ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ∥ 2 ≤ ( 30 d + 36 ) ∥ L 0 ϕ ∥ 2 + [ ( 30 d + 6 ) ( − α ) + + 3 γ 2 ] ∥ ∇ x ϕ ∥ 2 ≤ ( 66 d + ( 30 d + 6 ) ( − α ) + + 3 γ 2 a ) D a ( f S ) , (58) \begin{aligned}
\lVert(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rVert^2
&\le(30d+36)\lVert\mathcal{L}_0\phi\rVert^2\\
&\quad+[(30d+6)(-\alpha)_++3\gamma^2]\lVert\nabla_x\phi\rVert^2\\
&\le\left(66d+\frac{(30d+6)(-\alpha)_++3\gamma^2}{a}\right)\mathcal{D}_a(f_S), \tag*{(58)}
\end{aligned} ∥( I − Π v ) L ( v ⋅ ∇ x ϕ ) ∥ 2 ≤ ( 30 d + 36 ) ∥ L 0 ϕ ∥ 2 + [( 30 d + 6 ) ( − α ) + + 3 γ 2 ] ∥ ∇ x ϕ ∥ 2 ≤ ( 66 d + a ( 30 d + 6 ) ( − α ) + + 3 γ 2 ) D a ( f S ) , (58) using d ≥ 1 d\ge1 d ≥ 1
For (35) , assume also β < ∞ \beta<\infty β < ∞ ∣ ∇ U ⋅ ∇ x ϕ ∣ 2 ≤ ∣ ∇ U ∣ 2 ∣ ∇ x ϕ ∣ 2 \lvert\nabla U\cdot\nabla_x\phi\rvert^2\le\lvert\nabla U\rvert^2\lvert\nabla_x\phi\rvert^2 ∣ ∇ U ⋅ ∇ x ϕ ∣ 2 ≤ ∣ ∇ U ∣ 2 ∣ ∇ x ϕ ∣ 2 (51) with u = ∂ x j ϕ u=\partial_{x_j}\phi u = ∂ x j ϕ i , j i,j i , j (50) :
∥ ∇ U ⋅ ∇ x ϕ ∥ 2 ≤ ∑ i , j = 1 d ∥ ( ∂ x i U ) ( ∂ x j ϕ ) ∥ 2 ≤ 2 β d ∥ ∇ x ϕ ∥ 2 + 4 ∥ ∇ x 2 ϕ ∥ 2 ≤ 4 ∥ L 0 ϕ ∥ 2 + [ 2 β d + 4 ( − α ) + ] ∥ ∇ x ϕ ∥ 2 . (59) \begin{aligned}
\lVert\nabla U\cdot\nabla_x\phi\rVert^2
&\le\sum_{i,j=1}^{d}\lVert(\partial_{x_i}U)(\partial_{x_j}\phi)\rVert^2\\
&\le2\beta d\lVert\nabla_x\phi\rVert^2+4\lVert\nabla_x^2\phi\rVert^2\\
&\le4\lVert\mathcal{L}_0\phi\rVert^2+[2\beta d+4(-\alpha)_+]\lVert\nabla_x\phi\rVert^2. \tag*{(59)}
\end{aligned} ∥ ∇ U ⋅ ∇ x ϕ ∥ 2 ≤ i , j = 1 ∑ d ∥( ∂ x i U ) ( ∂ x j ϕ ) ∥ 2 ≤ 2 β d ∥ ∇ x ϕ ∥ 2 + 4 ∥ ∇ x 2 ϕ ∥ 2 ≤ 4 ∥ L 0 ϕ ∥ 2 + [ 2 β d + 4 ( − α ) + ] ∥ ∇ x ϕ ∥ 2 . (59) This time (56) and (38) give
∥ ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ∥ 2 ≤ 66 ∥ L 0 ϕ ∥ 2 + [ 66 ( − α ) + + 30 β d + 3 γ 2 ] ∥ ∇ x ϕ ∥ 2 ≤ ( 66 + 66 ( − α ) + + 30 β d + 3 γ 2 a ) D a ( f S ) . (60) \begin{aligned}
\lVert(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\rVert^2
&\le66\lVert\mathcal{L}_0\phi\rVert^2\\
&\quad+[66(-\alpha)_++30\beta d+3\gamma^2]\lVert\nabla_x\phi\rVert^2\\
&\le\left(66+\frac{66(-\alpha)_++30\beta d+3\gamma^2}{a}\right)\mathcal{D}_a(f_S). \tag*{(60)}
\end{aligned} ∥( I − Π v ) L ( v ⋅ ∇ x ϕ ) ∥ 2 ≤ 66 ∥ L 0 ϕ ∥ 2 + [ 66 ( − α ) + + 30 β d + 3 γ 2 ] ∥ ∇ x ϕ ∥ 2 ≤ ( 66 + a 66 ( − α ) + + 30 β d + 3 γ 2 ) D a ( f S ) . (60) Thus Cauchy–Schwarz gives the cross-term estimates (34) and (35) . □
Estimates for the Zigzag sampler.
Proof of Lemma 3.3, ZZ part. For a coordinate flip,
( F i v ) ⋅ ∇ x ϕ − v ⋅ ∇ x ϕ = − 2 v i ∂ x i ϕ . (F_i v)\cdot\nabla_x\phi-v\cdot\nabla_x\phi=-2v_i\partial_{x_i}\phi. ( F i v ) ⋅ ∇ x ϕ − v ⋅ ∇ x ϕ = − 2 v i ∂ x i ϕ . Consequently,
L ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v − 2 ∑ i = 1 d ( v i ∂ x i U ) + v i ∂ x i ϕ − γ v ⋅ ∇ x ϕ . (61) \mathcal{L}(v\cdot\nabla_x\phi)=v^\mathsf{T}\nabla_x^2\phi\,v-2\sum_{i=1}^{d}(v_i\partial_{x_i}U)_+v_i\partial_{x_i}\phi-\gamma v\cdot\nabla_x\phi. \tag*{(61)} L ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v − 2 i = 1 ∑ d ( v i ∂ x i U ) + v i ∂ x i ϕ − γ v ⋅ ∇ x ϕ . (61) Evenness of the one-dimensional Gaussian gives, for each fixed x x x
∫ 2 v i ( v i ∂ x i U ) + d μ v = ∂ x i U , \int2v_i(v_i\partial_{x_i}U)_+\,\mathrm{d}\mu_v=\partial_{x_i}U, ∫ 2 v i ( v i ∂ x i U ) + d μ v = ∂ x i U , ∫ [ 2 v i ( v i ∂ x i U ) + ] 2 d μ v = 6 ( ∂ x i U ) 2 . \int[2v_i(v_i\partial_{x_i}U)_+]^2\,\mathrm{d}\mu_v=6(\partial_{x_i}U)^2. ∫ [ 2 v i ( v i ∂ x i U ) + ] 2 d μ v = 6 ( ∂ x i U ) 2 . Both identities also hold when ∂ x i U = 0 \partial_{x_i}U=0 ∂ x i U = 0 (61) is therefore Δ x ϕ − ∇ U ⋅ ∇ x ϕ \Delta_x\phi-\nabla U\cdot\nabla_x\phi Δ x ϕ − ∇ U ⋅ ∇ x ϕ (32) for ZZ. Subtracting this average yields
( I − Π v ) L ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v − Δ x ϕ − ∑ i = 1 d [ 2 v i ( v i ∂ x i U ) + − ∂ x i U ] ∂ x i ϕ − γ v ⋅ ∇ x ϕ . (62) (I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)=v^{\mathsf{T}}\nabla_x^2\phi\,v-\Delta_x\phi-\sum_{i=1}^{d}\left[2v_i(v_i\partial_{x_i}U)_+-\partial_{x_i}U\right]\partial_{x_i}\phi-\gamma v\cdot\nabla_x\phi. \tag*{(62)} ( I − Π v ) L ( v ⋅ ∇ x ϕ ) = v T ∇ x 2 ϕ v − Δ x ϕ − i = 1 ∑ d [ 2 v i ( v i ∂ x i U ) + − ∂ x i U ] ∂ x i ϕ − γ v ⋅ ∇ x ϕ . (62) The summands in the second term have mean zero under μ v \mu_v μ v
∫ { ∑ i = 1 d [ 2 v i ( v i ∂ x i U ) + − ∂ x i U ] ∂ x i ϕ } 2 d μ v = 5 ∑ i = 1 d ( ∂ x i U ) 2 ( ∂ x i ϕ ) 2 . \int\left\{\sum_{i=1}^{d}\left[2v_i(v_i\partial_{x_i}U)_+-\partial_{x_i}U\right]\partial_{x_i}\phi\right\}^{2}\,\mathrm{d}\mu_v
=5\sum_{i=1}^{d}(\partial_{x_i}U)^2(\partial_{x_i}\phi)^2. ∫ { i = 1 ∑ d [ 2 v i ( v i ∂ x i U ) + − ∂ x i U ] ∂ x i ϕ } 2 d μ v = 5 i = 1 ∑ d ( ∂ x i U ) 2 ( ∂ x i ϕ ) 2 . Using the quadratic Gaussian moment in Section 3.3.1 and ( r + s + t ) 2 ≤ 3 ( r 2 + s 2 + t 2 ) (r+s+t)^2\leq3(r^2+s^2+t^2) ( r + s + t ) 2 ≤ 3 ( r 2 + s 2 + t 2 ) (62) , we obtain
∥ ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ∥ 2 ≤ 6 ∥ ∇ x 2 ϕ ∥ 2 + 15 ∑ i = 1 d ∥ ( ∂ x i U ) ( ∂ x i ϕ ) ∥ 2 + 3 γ 2 ∥ ∇ x ϕ ∥ 2 . (63) \left\|(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\right\|^2\leq6\left\|\nabla_x^2\phi\right\|^2+15\sum_{i=1}^{d}\left\|(\partial_{x_i}U)(\partial_{x_i}\phi)\right\|^2+3\gamma^2\left\|\nabla_x\phi\right\|^2. \tag*{(63)} ∥ ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ∥ 2 ≤ 6 ∇ x 2 ϕ 2 + 15 i = 1 ∑ d ∥ ( ∂ x i U ) ( ∂ x i ϕ ) ∥ 2 + 3 γ 2 ∥ ∇ x ϕ ∥ 2 . (63) Here (51) is needed only with u = ∂ x i ϕ u=\partial_{x_i}\phi u = ∂ x i ϕ i i i i , j i,j i , j
∑ i = 1 d ∥ ( ∂ x i U ) ( ∂ x i ϕ ) ∥ 2 ≤ 2 β ∥ ∇ x ϕ ∥ 2 + 4 ∑ i = 1 d ∥ ∂ x i x i ϕ ∥ 2 \sum_{i=1}^{d}\left\|(\partial_{x_i}U)(\partial_{x_i}\phi)\right\|^2\leq2\beta\left\|\nabla_x\phi\right\|^2+4\sum_{i=1}^{d}\left\|\partial_{x_i x_i}\phi\right\|^2 i = 1 ∑ d ∥ ( ∂ x i U ) ( ∂ x i ϕ ) ∥ 2 ≤ 2 β ∥ ∇ x ϕ ∥ 2 + 4 i = 1 ∑ d ∥ ∂ x i x i ϕ ∥ 2 ≤ 4 ∥ L 0 ϕ ∥ 2 + [ 2 β + 4 ( − α ) + ] ∥ ∇ x ϕ ∥ 2 . (64) \leq4\left\|\mathcal{L}_0\phi\right\|^2+\left[2\beta+4(-\alpha)_+\right]\left\|\nabla_x\phi\right\|^2. \tag*{(64)} ≤ 4 ∥ L 0 ϕ ∥ 2 + [ 2 β + 4 ( − α ) + ] ∥ ∇ x ϕ ∥ 2 . (64) The last step uses ∑ i ∥ ∂ x i x i ϕ ∥ 2 ≤ ∥ ∇ x 2 ϕ ∥ 2 \sum_i\left\|\partial_{x_i x_i}\phi\right\|^2\leq\left\|\nabla_x^2\phi\right\|^2 ∑ i ∥ ∂ x i x i ϕ ∥ 2 ≤ ∇ x 2 ϕ 2 (50) . Combining (63) , (64) , and (38) , we conclude that
∥ ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ∥ 2 ≤ 66 ∥ L 0 ϕ ∥ 2 + [ 66 ( − α ) + + 30 β + 3 γ 2 ] ∥ ∇ x ϕ ∥ 2 ≤ ( 66 + 66 ( − α ) + + 30 β + 3 γ 2 a ) D a ( f S ) . (65) \begin{aligned}
\left\|(I-\Pi_v)\mathcal{L}(v\cdot\nabla_x\phi)\right\|^2\leq66\left\|\mathcal{L}_0\phi\right\|^2+\left[66(-\alpha)_++30\beta+3\gamma^2\right]\left\|\nabla_x\phi\right\|^2 \\
&\leq\left(66+\frac{66(-\alpha)_++30\beta+3\gamma^2}{a}\right)\mathcal{D}_a(f_S). \tag*{(65)}
\end{aligned} ∥ ( I − Π v ) L ( v ⋅ ∇ x ϕ ) ∥ 2 ≤ 66 ∥ L 0 ϕ ∥ 2 + [ 66 ( − α ) + + 30 β + 3 γ 2 ] ∥ ∇ x ϕ ∥ 2 ≤ ( 66 + a 66 ( − α ) + + 30 β + 3 γ 2 ) D a ( f S ) . (65) Hence Cauchy–Schwarz gives (36) .
Convergence and mixing-time bounds
We prove the decay estimate for the Rényi-2 divergence and the resulting mixing-time bounds.
Proof of Theorem 1.1 . We apply Proposition 3.4 to the spatial component f S , t f_{S,t} f S , t L 0 2 ( μ x ) L^2_0(\mu_x) L 0 2 ( μ x ) f S , t ≥ − 1 f_{S,t}\geq-1 f S , t ≥ − 1 L 2 ( μ ) L^2(\mu) L 2 ( μ ) P t ∗ \mathcal{P}_t^* P t ∗
0 ≤ log ( 1 + ∥ f S ∥ 2 ) ≤ log ( 1 + ∥ f t ∥ 2 ) ≤ log ( 1 + ∥ f 0 ∥ 2 ) ≤ R . 0\leq\log(1+\|f_S\|^2)\leq\log(1+\|f_t\|^2)\leq\log(1+\|f_0\|^2)\leq R. 0 ≤ log ( 1 + ∥ f S ∥ 2 ) ≤ log ( 1 + ∥ f t ∥ 2 ) ≤ log ( 1 + ∥ f 0 ∥ 2 ) ≤ R . Choose a = C L S I ( 1 + R ) a=C_{\mathrm{LSI}}(1+R) a = C LSI ( 1 + R ) Proposition 3.4 uniformly in time:
2 a + C L S I ( 1 + log ( 1 + ∥ f S ∥ 2 ) ) ≤ 2 a + C L S I ( 1 + R ) = 3 a . 2a+C_{\mathrm{LSI}}(1+\log(1+\lVert f_S\rVert^2))\le2a+C_{\mathrm{LSI}}(1+R)=3a. 2 a + C LSI ( 1 + log ( 1 + ∥ f S ∥ 2 )) ≤ 2 a + C LSI ( 1 + R ) = 3 a . Thus, at every time,
D a ( f S ) ≥ C L S I 3 a ( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) . (66) \mathcal{D}_a(f_S)\ge\frac{C_{\mathrm{LSI}}}{3a}(1+\lVert f_S\rVert^2)\log(1+\lVert f_S\rVert^2). \tag*{(66)} D a ( f S ) ≥ 3 a C LSI ( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) . (66) We next combine the macroscopic and velocity dissipation into
d d t L a , ϵ ( f t ) ≤ − C L S I ϵ 6 a ( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) − γ 2 ∥ f F ∥ 2 ≤ − C L S I ϵ 6 a [ ( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) + ( 1 + R ) ∥ f F ∥ 2 ] ≤ − C L S I ϵ 6 a ( 1 + ∥ f t ∥ 2 ) log ( 1 + ∥ f t ∥ 2 ) . (67) \begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{L}_{a,\epsilon}(f_t)
&\le-\frac{C_{\mathrm{LSI}}\epsilon}{6a}(1+\lVert f_S\rVert^2)\log(1+\lVert f_S\rVert^2)-\frac{\gamma}{2}\lVert f_F\rVert^2\\
&\le-\frac{C_{\mathrm{LSI}}\epsilon}{6a}\left[(1+\lVert f_S\rVert^2)\log(1+\lVert f_S\rVert^2)+(1+R)\lVert f_F\rVert^2\right]\\
&\le-\frac{C_{\mathrm{LSI}}\epsilon}{6a}(1+\lVert f_t\rVert^2)\log(1+\lVert f_t\rVert^2). \tag*{(67)}
\end{aligned} d t d L a , ϵ ( f t ) ≤ − 6 a C LSI ϵ ( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) − 2 γ ∥ f F ∥ 2 ≤ − 6 a C LSI ϵ [ ( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) + ( 1 + R ) ∥ f F ∥ 2 ] ≤ − 6 a C LSI ϵ ( 1 + ∥ f t ∥ 2 ) log ( 1 + ∥ f t ∥ 2 ) . (67) The first inequality follows from Proposition 3.1 and (66) . The second holds because a = C L S I ( 1 + R ) a=C_{\mathrm{LSI}}(1+R) a = C LSI ( 1 + R ) ϵ ≤ γ \epsilon\le\gamma ϵ ≤ γ
C L S I ϵ 6 a ( 1 + R ) = ϵ 6 ≤ γ 2 . \frac{C_{\mathrm{LSI}}\epsilon}{6a}(1+R)=\frac{\epsilon}{6}\le\frac{\gamma}{2}. 6 a C LSI ϵ ( 1 + R ) = 6 ϵ ≤ 2 γ . For the last inequality, orthogonality gives ∥ f ∥ 2 − ∥ f S ∥ 2 = ∥ f F ∥ 2 \lVert f\rVert^2-\lVert f_S\rVert^2=\lVert f_F\rVert^2 ∥ f ∥ 2 − ∥ f S ∥ 2 = ∥ f F ∥ 2 1 + ∥ f ∥ 2 1+\lVert f\rVert^2 1 + ∥ f ∥ 2 1 + ∥ f S ∥ 2 1+\lVert f_S\rVert^2 1 + ∥ f S ∥ 2 R R R
( 1 + ∥ f ∥ 2 ) log ( 1 + ∥ f ∥ 2 ) − ( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) = ∫ 1 + ∥ f S ∥ 2 1 + ∥ f ∥ 2 ( 1 + log s ) d s ≤ ( 1 + R ) ( ∥ f ∥ 2 − ∥ f S ∥ 2 ) = ( 1 + R ) ∥ f F ∥ 2 . \begin{aligned}
&(1+\lVert f\rVert^2)\log(1+\lVert f\rVert^2)-(1+\lVert f_S\rVert^2)\log(1+\lVert f_S\rVert^2)\\
&=\int_{1+\lVert f_S\rVert^2}^{1+\lVert f\rVert^2}(1+\log s)\,\mathrm{d}s\\
&\le(1+R)(\lVert f\rVert^2-\lVert f_S\rVert^2)=(1+R)\lVert f_F\rVert^2.
\end{aligned} ( 1 + ∥ f ∥ 2 ) log ( 1 + ∥ f ∥ 2 ) − ( 1 + ∥ f S ∥ 2 ) log ( 1 + ∥ f S ∥ 2 ) = ∫ 1 + ∥ f S ∥ 2 1 + ∥ f ∥ 2 ( 1 + log s ) d s ≤ ( 1 + R ) (∥ f ∥ 2 − ∥ f S ∥ 2 ) = ( 1 + R ) ∥ f F ∥ 2 . It remains to turn (67) into decay of R 2 ( ϱ t ∣ μ ) = log ( 1 + ∥ f t ∥ 2 ) R_2(\varrho_t\mid\mu)=\log(1+\lVert f_t\rVert^2) R 2 ( ϱ t ∣ μ ) = log ( 1 + ∥ f t ∥ 2 ) (67) equals − C L S I ϵ 6 a ( 1 + ∥ f t ∥ 2 ) R 2 ( ϱ t ∣ μ ) -\frac{C_{\mathrm{LSI}}\epsilon}{6a}(1+\lVert f_t\rVert^2)R_2(\varrho_t\mid\mu) − 6 a C LSI ϵ ( 1 + ∥ f t ∥ 2 ) R 2 ( ϱ t ∣ μ ) ∥ f t ∥ 2 \lVert f_t\rVert^2 ∥ f t ∥ 2 ∥ f t ∥ 2 \lVert f_t\rVert^2 ∥ f t ∥ 2 R 2 R_2 R 2
H ( t ) = log ( 1 + 4 3 L a , ϵ ( f t ) ) . H(t)=\log\left(1+\frac{4}{3}\mathcal{L}_{a,\epsilon}(f_t)\right). H ( t ) = log ( 1 + 3 4 L a , ϵ ( f t ) ) . The factor 4 / 3 4/3 4/3 Lemma 2.3 gives
1 3 ∥ f t ∥ 2 ≤ 4 3 L a , ϵ ( f t ) ≤ ∥ f t ∥ 2 . \frac{1}{3}\lVert f_t\rVert^2\le\frac{4}{3}\mathcal{L}_{a,\epsilon}(f_t)\le\lVert f_t\rVert^2. 3 1 ∥ f t ∥ 2 ≤ 3 4 L a , ϵ ( f t ) ≤ ∥ f t ∥ 2 . Since log ( 1 + x / 3 ) ≥ 1 3 log ( 1 + x ) \log(1+x/3)\ge\frac{1}{3}\log(1+x) log ( 1 + x /3 ) ≥ 3 1 log ( 1 + x ) x ≥ 0 x\ge0 x ≥ 0
1 3 R 2 ( ϱ t ∣ μ ) ≤ H ( t ) ≤ R 2 ( ϱ t ∣ μ ) . (68) \frac{1}{3}R_2(\varrho_t\mid\mu)\le H(t)\le R_2(\varrho_t\mid\mu). \tag*{(68)} 3 1 R 2 ( ϱ t ∣ μ ) ≤ H ( t ) ≤ R 2 ( ϱ t ∣ μ ) . (68) By (67) ,
H ′ ( t ) = 4 3 d d t L a , ϵ ( f t ) 1 + 4 3 L a , ϵ ( f t ) ≤ − 2 C L S I ϵ 9 a 1 + ∥ f t ∥ 2 1 + 4 3 L a , ϵ ( f t ) R 2 ( ϱ t ∣ μ ) . H'(t)=\frac{\frac{4}{3}\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{L}_{a,\epsilon}(f_t)}{1+\frac{4}{3}\mathcal{L}_{a,\epsilon}(f_t)}
\le-\frac{2C_{\mathrm{LSI}}\epsilon}{9a}\frac{1+\lVert f_t\rVert^2}{1+\frac{4}{3}\mathcal{L}_{a,\epsilon}(f_t)}R_2(\varrho_t\mid\mu). H ′ ( t ) = 1 + 3 4 L a , ϵ ( f t ) 3 4 d t d L a , ϵ ( f t ) ≤ − 9 a 2 C LSI ϵ 1 + 3 4 L a , ϵ ( f t ) 1 + ∥ f t ∥ 2 R 2 ( ϱ t ∣ μ ) . The fraction ( 1 + ∥ f t ∥ 2 ) / ( 1 + 4 3 L a , ϵ ( f t ) ) (1+\lVert f_t\rVert^2)/(1+\frac{4}{3}\mathcal{L}_{a,\epsilon}(f_t)) ( 1 + ∥ f t ∥ 2 ) / ( 1 + 3 4 L a , ϵ ( f t )) 4 3 L a , ϵ ( f t ) \frac{4}{3}\mathcal{L}_{a,\epsilon}(f_t) 3 4 L a , ϵ ( f t ) R 2 ( ϱ t ∣ μ ) ≥ H ( t ) R_2(\varrho_t\mid\mu)\ge H(t) R 2 ( ϱ t ∣ μ ) ≥ H ( t ) (68) . Hence
H ′ ( t ) ≤ − 2 C L S I ϵ 9 a H ( t ) . (69) H'(t)\le-\frac{2C_{\mathrm{LSI}}\epsilon}{9a}H(t). \tag*{(69)} H ′ ( t ) ≤ − 9 a 2 C LSI ϵ H ( t ) . (69) Grönwall’s inequality for (69) , together with (68) , gives
R 2 ( ϱ t ∣ μ ) ≤ 3 H ( t ) ≤ 3 e − 2 C L S I ϵ t / ( 9 a ) H ( 0 ) ≤ 3 e − 2 C L S I ϵ t / ( 9 a ) R 2 ( ϱ 0 ∣ μ ) . R_2(\varrho_t \mid\mu) \le3H(t) \le3e^{-2C_{\mathrm{LSI}}\epsilon t/(9a)}H(0) \le3e^{-2C_{\mathrm{LSI}}\epsilon t/(9a)}R_2(\varrho_0 \mid\mu). R 2 ( ϱ t ∣ μ ) ≤ 3 H ( t ) ≤ 3 e − 2 C LSI ϵ t / ( 9 a ) H ( 0 ) ≤ 3 e − 2 C LSI ϵ t / ( 9 a ) R 2 ( ϱ 0 ∣ μ ) . This is (8) with Λ = 2 C L S I ϵ / ( 9 a ) \Lambda= 2C_{\mathrm{LSI}}\epsilon/(9a) Λ = 2 C LSI ϵ / ( 9 a ) (28) and a = C L S I ( 1 + R ) a = C_{\mathrm{LSI}}(1+R) a = C LSI ( 1 + R ) (9) –(11) . This proves the theorem when f 0 ∈ D ( L ∗ ) f_0 \in D(\mathcal{L}^*) f 0 ∈ D ( L ∗ ) Proposition 3.1 . Standard approximation and the L 2 ( μ ) L^2(\mu) L 2 ( μ ) P t ∗ \mathcal{P}_t^* P t ∗
The choice of the shift drives the improved dependence on R R R C L S I ϵ C_{\mathrm{LSI}}\epsilon C LSI ϵ Proposition 3.4 , which is at most 2 a + C L S I ( 1 + R ) 2a+C_{\mathrm{LSI}}(1+R) 2 a + C LSI ( 1 + R ) 1 / ( 2 a ) 1/(2\sqrt{a}) 1/ ( 2 a ) ϵ ≤ a \epsilon\le\sqrt{a} ϵ ≤ a Lemma 2.3 ). For fixed curvature and dimension, (28) shows that a friction or refreshment rate γ \gamma γ a \sqrt{a} a a \sqrt{a} a C L S I a / ( a + C L S I R ) C_{\mathrm{LSI}}\sqrt{a}/(a+C_{\mathrm{LSI}}R) C LSI a / ( a + C LSI R ) R ≥ 1 R \ge1 R ≥ 1 (44) , the factor C L S I / ( a + C L S I ) C_{\mathrm{LSI}}/(a+C_{\mathrm{LSI}}) C LSI / ( a + C LSI ) a a a a = 0 a=0 a = 0 [20 Corollary 2] . With a a a C L S I C_{\mathrm{LSI}} C LSI C L S I / R \sqrt{C_{\mathrm{LSI}}/R} C LSI / R a = C L S I ( 1 + R ) a=C_{\mathrm{LSI}}(1+R) a = C LSI ( 1 + R ) C L S I / R \sqrt{C_{\mathrm{LSI}}/R} C LSI / R 3 C L S I ( 1 + R ) 3C_{\mathrm{LSI}}(1+R) 3 C LSI ( 1 + R ) C L S I ( 1 + R ) C_{\mathrm{LSI}}(1+R) C LSI ( 1 + R )
Proof of Corollary 1.2. We first prove the Rényi-2 bound, with γ \gamma γ (12) . We apply Theorem 1.1 in stages with successively halved divergence bounds, keeping γ \gamma γ r r r Theorem 1.1 as
Λ ( r ) = C L S I γ c [ γ 2 + B ( r ) ] . \Lambda(r)=\frac{C_{\mathrm{LSI}}\gamma}{c[\gamma^2+B(r)]}. Λ ( r ) = c [ γ 2 + B ( r )] C LSI γ . where c = 9 / 2 c=9/2 c = 9/2 c = 450 c=450 c = 450 B ( r ) B(r) B ( r ) (9) –(11) independent of γ 2 \gamma^2 γ 2 r r r (12) is precisely
γ 2 = B ( R / L ) . \gamma^2=B(R/\mathcal{L}). γ 2 = B ( R / L ) . Set
N = ⌈ log 2 R η ⌉ , R j = 2 − j R ( 0 ≤ j ≤ N ) , t j = log 6 Λ ( R j ) ( 0 ≤ j < N ) . N=\left\lceil\log_2\frac{R}{\eta}\right\rceil,\qquad R_j=2^{-j}R\quad(0\le j\le N),\qquad t_j=\frac{\log6}{\Lambda(R_j)}\quad(0\le j<N). N = ⌈ log 2 η R ⌉ , R j = 2 − j R ( 0 ≤ j ≤ N ) , t j = Λ ( R j ) log 6 ( 0 ≤ j < N ) . If the divergence at the start of stage j j j R j R_j R j Theorem 1.1 bounds it at the end by
3 e − Λ ( R j ) t j R j = R j + 1 . 3e^{-\Lambda(R_j)t_j}R_j=R_{j+1}. 3 e − Λ ( R j ) t j R j = R j + 1 . The theorem applies at each endpoint, including when R j < 1 R_j<1 R j < 1 R N ≤ η R_N\le\eta R N ≤ η ∑ j < N t j \sum_{j<N}t_j ∑ j < N t j ∥ f t ∥ \lVert f_t\rVert ∥ f t ∥
Since R / η ≥ 2 R/\eta\ge2 R / η ≥ 2
L log 6 ≤ N ≤ L log 2 , ∑ j = 0 N − 1 R j ≤ 2 R \frac{\mathcal{L}}{\log6}\le N\le\frac{\mathcal{L}}{\log2},\qquad\sum_{j=0}^{N-1}R_j\le2R log 6 L ≤ N ≤ log 2 L , j = 0 ∑ N − 1 R j ≤ 2 R imply, for every affine function b ( r ) = b 0 + b 1 r b(r)=b_0+b_1r b ( r ) = b 0 + b 1 r b 0 , b 1 ≥ 0 b_0,b_1\ge0 b 0 , b 1 ≥ 0
∑ j = 0 N − 1 b ( R j ) ≤ N b 0 + 2 b 1 R ≤ 4 N b ( R / L ) . \sum_{j=0}^{N-1}b(R_j)\le Nb_0+2b_1R\le4N b(R/\mathcal{L}). j = 0 ∑ N − 1 b ( R j ) ≤ N b 0 + 2 b 1 R ≤ 4 N b ( R / L ) . For BPS, apply this bound to each finite branch and take their minimum. In all three cases,
∑ j = 0 N − 1 B ( R j ) ≤ 4 N B ( R / L ) = 4 N γ 2 . \sum_{j=0}^{N-1} B(R_j) \le4N B(R/\mathcal{L}) = 4N\gamma^2. j = 0 ∑ N − 1 B ( R j ) ≤ 4 N B ( R / L ) = 4 N γ 2 . It follows that
∑ j = 0 N − 1 t j ≤ c log 6 C L S I γ [ N γ 2 + ∑ j = 0 N − 1 B ( R j ) ] ≤ 5 c log 6 log 2 L γ C L S I . \sum_{j=0}^{N-1} t_j \le\frac{c\log6}{C_{\mathrm{LSI}}\gamma}\left[N\gamma^2+\sum_{j=0}^{N-1}B(R_j)\right]
\le\frac{5c\log6}{\log2}\frac{\mathcal{L}\gamma}{C_{\mathrm{LSI}}}. j = 0 ∑ N − 1 t j ≤ C LSI γ c log 6 [ N γ 2 + j = 0 ∑ N − 1 B ( R j ) ] ≤ log 2 5 c log 6 C LSI L γ . The prefactor is below 60 for ULD and 6000 for BPS and ZZ. Substituting (12) proves the Rényi-2 bound.
For the total variation bound, η = 2 δ 2 \eta=2\delta^2 η = 2 δ 2 ( 0 , 1 / 2 ] (0,1/2] ( 0 , 1/2 ] 0 < δ ≤ 1 / 2 0<\delta\le1/2 0 < δ ≤ 1/2 (7) gives TV ( ϱ t , μ ) ≤ δ \operatorname{TV}(\varrho_t,\mu)\le\delta TV ( ϱ t , μ ) ≤ δ R 2 ( ϱ t ∣ μ ) ≤ 2 δ 2 R_2(\varrho_t\mid\mu)\le2\delta^2 R 2 ( ϱ t ∣ μ ) ≤ 2 δ 2
Sharpness for the Rényi-2 divergence
This appendix proves Theorem 1.3 , which shows that the order R log R \sqrt{R\log R} R log R
Proof of Theorem 1.3 . We construct one initial law, independent of γ \gamma γ γ > 0 \gamma>0 γ > 0 R 2 ( ϱ t ∣ μ ) ≤ 1 / 2 R_2(\varrho_t\mid\mu)\le1/2 R 2 ( ϱ t ∣ μ ) ≤ 1/2 t ≥ 0 t\ge0 t ≥ 0 t t t R / γ R/\gamma R / γ γ log R / α \gamma\log R/\alpha γ log R / α γ \gamma γ
Initial law. For U ( x ) = α x 2 / 2 U(x)=\alpha x^2/2 U ( x ) = α x 2 /2 v v v − v -v − v ( α x v ) + (\alpha xv)_+ ( α xv ) +
y 0 = R / 2 , p = e − R / 2 , y_0=\sqrt{R/2}, \qquad p=e^{-R/2}, y 0 = R /2 , p = e − R /2 , and choose b > 0 b>0 b > 0 μ v ( [ − b , b ] ) = p \mu_v([-b,b])=p μ v ([ − b , b ]) = p
α X 0 ∼ N ( y 0 , 1 ) , V 0 ∼ μ v ( ⋅ ∣ ∣ V ∣ ≤ b ) . \sqrt{\alpha}X_0\sim N(y_0,1), \qquad V_0\sim\mu_v(\mathord{\cdot}\mid|V|\le b). α X 0 ∼ N ( y 0 , 1 ) , V 0 ∼ μ v ( ⋅ ∣ ∣ V ∣ ≤ b ) . The translated spatial Gaussian has Rényi-2 divergence y 0 2 = R / 2 y_0^2=R/2 y 0 2 = R /2 − log p = R / 2 -\log p=R/2 − log p = R /2 R 2 ( ϱ 0 ∣ μ ) = R R_2(\varrho_0\mid\mu)=R R 2 ( ϱ 0 ∣ μ ) = R E V 0 = 0 \mathbb{E}V_0=0 E V 0 = 0 [ − b , b ] [-b,b] [ − b , b ] E V 0 2 ≤ 1 \mathbb{E}V_0^2\le1 E V 0 2 ≤ 1
Speed bound. Bounces and flips preserve ∣ V ∣ |V| ∣ V ∣ ∣ V t ∣ |V_t| ∣ V t ∣ e − γ t e^{-\gamma t} e − γ t 1 − e − γ t 1-e^{-\gamma t} 1 − e − γ t
1 + e − γ t ( p − 1 1 [ 0 , b ] − 1 ) . 1+e^{-\gamma t}\left(p^{-1}\mathbf{1}_{[0,b]}-1\right). 1 + e − γ t ( p − 1 1 [ 0 , b ] − 1 ) . Data processing for the map ( x , v ) ↦ ∣ v ∣ (x,v)\mapsto|v| ( x , v ) ↦ ∣ v ∣
R 2 ( ϱ t ∣ μ ) ≥ log [ 1 + e − 2 γ t ( e R / 2 − 1 ) ] . (70) R_2(\varrho_t\mid\mu)\ge\log\left[1+e^{-2\gamma t}\left(e^{R/2}-1\right)\right]. \tag*{(70)} R 2 ( ϱ t ∣ μ ) ≥ log [ 1 + e − 2 γ t ( e R /2 − 1 ) ] . (70) Since R 2 ( ϱ t ∣ μ ) ≤ 1 / 2 \mathrm{R}_2(\varrho_t \mid\mu) \le1/2 R 2 ( ϱ t ∣ μ ) ≤ 1/2 R ≥ e 8 R \ge e^8 R ≥ e 8
t ≥ 1 2 γ log e R / 2 − 1 e 1 / 2 − 1 ≥ R 4 γ . (71) t \ge\frac{1}{2\gamma}\log\frac{e^{R/2}-1}{e^{1/2}-1} \ge\frac{R}{4\gamma}. \tag*{(71)} t ≥ 2 γ 1 log e 1/2 − 1 e R /2 − 1 ≥ 4 γ R . (71) The same mixture gives E V s 2 = 1 + e − γ s ( E V 0 2 − 1 ) ≤ 1 \mathbb{E}V_s^2=1+e^{-\gamma s}(\mathbb{E}V_0^2-1)\le1 E V s 2 = 1 + e − γ s ( E V 0 2 − 1 ) ≤ 1 s ≥ 0 s\ge0 s ≥ 0
Return-time bound. We measure the distance from the origin on a logarithmic scale by log ( 1 + α x 2 ) \log(1+\alpha x^2) log ( 1 + α x 2 ) γ \gamma γ
Φ γ ( x , v ) = log ( 1 + α x 2 ) + 2 α x v γ ( 1 + α x 2 ) . \Phi_\gamma(x,v)=\log(1+\alpha x^2)+\frac{2\alpha xv}{\gamma(1+\alpha x^2)}. Φ γ ( x , v ) = log ( 1 + α x 2 ) + γ ( 1 + α x 2 ) 2 α xv . Write y = α x y=\sqrt{\alpha}x y = α x
L Φ γ ( x , v ) = α γ v 2 [ 2 ( 1 − y 2 ) ( 1 + y 2 ) 2 − 4 y 2 1 + y 2 1 { y v > 0 } ] ≥ − 4 α γ v 2 . (72) \begin{aligned}
\mathcal{L}\Phi_\gamma(x,v)=\frac{\alpha}{\gamma}v^2\left[\frac{2(1-y^2)}{(1+y^2)^2}-\frac{4y^2}{1+y^2}\mathbf{1}_{\{yv>0\}}\right] \\
&\ge-\frac{4\alpha}{\gamma}v^2. \tag*{(72)}
\end{aligned} L Φ γ ( x , v ) = γ α v 2 [ ( 1 + y 2 ) 2 2 ( 1 − y 2 ) − 1 + y 2 4 y 2 1 { y v > 0 } ] ≥ − γ 4 α v 2 . (72) The inequality holds because adding 4 to the bracket and replacing the indicator by 1 gives ( 6 + 2 y 2 ) / ( 1 + y 2 ) 2 ≥ 0 (6+2y^2)/(1+y^2)^2\ge0 ( 6 + 2 y 2 ) / ( 1 + y 2 ) 2 ≥ 0 E V s 2 ≤ 1 \mathbb{E}V_s^2\le1 E V s 2 ≤ 1
E Φ γ ( X t , V t ) ≥ E Φ γ ( X 0 , V 0 ) − 4 α t γ . (73) \mathbb{E}\Phi_\gamma(X_t,V_t)\ge\mathbb{E}\Phi_\gamma(X_0,V_0)-\frac{4\alpha t}{\gamma}. \tag*{(73)} E Φ γ ( X t , V t ) ≥ E Φ γ ( X 0 , V 0 ) − γ 4 α t . (73) We first apply Dynkin’s formula with localization on a finite interval [ 0 , T ] [0,T] [ 0 , T ] γ T \gamma T γ T E sup s ≤ T ∣ V s ∣ 2 ≤ 1 + γ T \mathbb{E}\sup_{s\le T}|V_s|^2\le1+\gamma T E sup s ≤ T ∣ V s ∣ 2 ≤ 1 + γ T X s = X 0 + ∫ 0 s V r d r X_s=X_0+\int_0^sV_r\,\mathrm{d}r X s = X 0 + ∫ 0 s V r d r E sup s ≤ T ∣ X s ∣ 2 < ∞ \mathbb{E}\sup_{s\le T}|X_s|^2<\infty E sup s ≤ T ∣ X s ∣ 2 < ∞ Φ γ \Phi_\gamma Φ γ ∣ L Φ γ ∣ ≤ 4 α v 2 / γ |\mathcal{L}\Phi_\gamma|\le4\alpha v^2/\gamma ∣ L Φ γ ∣ ≤ 4 α v 2 / γ
It remains to compare the initial and terminal values of E Φ γ \mathbb{E}\Phi_\gamma E Φ γ Φ γ \Phi_\gamma Φ γ Z ∼ N ( 0 , 1 ) Z\sim N(0,1) Z ∼ N ( 0 , 1 ) 1 + y 0 2 ≤ 2 [ 1 + ( y 0 + Z ) 2 ] ( 1 + Z 2 ) 1+y_0^2\le2[1+(y_0+Z)^2](1+Z^2) 1 + y 0 2 ≤ 2 [ 1 + ( y 0 + Z ) 2 ] ( 1 + Z 2 )
E Φ γ ( X 0 , V 0 ) ≥ log ( 1 + R / 2 ) − 2 log 2. \mathbb{E}\Phi_\gamma(X_0,V_0)\ge\log(1+R/2)-2\log2. E Φ γ ( X 0 , V 0 ) ≥ log ( 1 + R /2 ) − 2 log 2. At equilibrium, Jensen’s inequality gives ∫ Φ γ d μ ≤ log 2 \int\Phi_\gamma\,\mathrm{d}\mu\le\log2 ∫ Φ γ d μ ≤ log 2 ∣ 2 Z / ( 1 + Z 2 ) ∣ ≤ 1 |2Z/(1+Z^2)|\le1 ∣2 Z / ( 1 + Z 2 ) ∣ ≤ 1 α / γ 2 \alpha/\gamma^2 α / γ 2
Var μ Φ γ ≤ 1 + α γ 2 . \operatorname{Var}_\mu\Phi_\gamma\le1+\frac{\alpha}{\gamma^2}. Var μ Φ γ ≤ 1 + γ 2 α . Put ζ = e 1 / 2 − 1 < 1 \zeta=\sqrt{e^{1/2}-1}<1 ζ = e 1/2 − 1 < 1 R 2 ( ϱ t ∣ μ ) ≤ 1 / 2 \mathrm{R}_2(\varrho_t\mid\mu)\le1/2 R 2 ( ϱ t ∣ μ ) ≤ 1/2 χ 2 ( ϱ t ∣ μ ) ≤ ζ 2 \chi^2(\varrho_t\mid\mu)\le\zeta^2 χ 2 ( ϱ t ∣ μ ) ≤ ζ 2
E Φ γ ( X t , V t ) ≤ log 2 + ζ 1 + α / γ 2 . \mathbb{E}\Phi_\gamma(X_t,V_t)\le\log2+\zeta\sqrt{1+\alpha/\gamma^2}. E Φ γ ( X t , V t ) ≤ log 2 + ζ 1 + α / γ 2 . Combining this with (A.4) and using 1 + α / γ 2 ≤ 1 + α / γ \sqrt{1+\alpha/\gamma^2}\le1+\sqrt{\alpha}/\gamma 1 + α / γ 2 ≤ 1 + α / γ
t + ζ 4 α ≥ γ 4 α [ log ( 1 + R / 2 ) − 3 log 2 − ζ ] . t+\frac{\zeta}{4\sqrt{\alpha}}\ge\frac{\gamma}{4\alpha}\left[\log(1+R/2)-3\log2-\zeta\right]. t + 4 α ζ ≥ 4 α γ [ log ( 1 + R /2 ) − 3 log 2 − ζ ] . Since log R ≥ 8 \log R\ge8 log R ≥ 8 4 log 2 + 1 < 4 4\log2+1<4 4 log 2 + 1 < 4 1 2 log R \frac{1}{2}\log R 2 1 log R
Conclusion. Combining the last display with (A.2) gives
t + 1 4 α ≥ max { R 4 γ , γ log R 8 α } ≥ 1 α R log R 32 . t+\frac{1}{4\sqrt{\alpha}}\ge\max\left\{\frac{R}{4\gamma},\frac{\gamma\log R}{8\alpha}\right\}\ge\frac{1}{\sqrt{\alpha}}\sqrt{\frac{R\log R}{32}}. t + 4 α 1 ≥ max { 4 γ R , 8 α γ log R } ≥ α 1 32 R log R . Hence t ≥ ( 16 α ) − 1 R log R t\ge(16\sqrt{\alpha})^{-1}\sqrt{R\log R} t ≥ ( 16 α ) − 1 R log R R ≥ e 8 R\ge e^{8} R ≥ e 8 γ > 0 \gamma>0 γ > 0 γ \gamma γ (14) . □
Appendix B. Lower bounds for the Zigzag sampler
The ZZ bound in Corollary 1.2 depends on the initial divergence through C L S I − 1 / 2 R C_{\mathrm{LSI}}^{-1/2}\sqrt{\mathcal{R}} C LSI − 1/2 R C L S I − 1 β L C_{\mathrm{LSI}}^{-1}\sqrt{\beta}\mathcal{L} C LSI − 1 β L d ≥ 2 d\ge2 d ≥ 2 d ≥ 2 d\ge2 d ≥ 2 β \beta β β \beta β (10) also bounds ZZ. The first example is an isotropic Gaussian target with an initial law displaced along a coordinate axis. The second is a family of two-dimensional Gaussian targets confined in a direction oblique to the coordinate axes; it also shows that strong convexity alone gives no uniform ZZ bound, in contrast with BPS.
Theorem B.1 (Dependence on the initial divergence). Let d ≥ 1 d\ge1 d ≥ 1 α > 0 \alpha>0 α > 0 U ( x ) = α ∣ x ∣ 2 / 2 U(x)=\alpha|x|^{2}/2 U ( x ) = α ∣ x ∣ 2 /2 R d \mathbb{R}^{d} R d μ = N ( 0 , α − 1 I d ) ⊗ μ v \mu=N(0,\alpha^{-1}I_{d})\otimes\mu_{v} μ = N ( 0 , α − 1 I d ) ⊗ μ v (2) holds with C L S I = α C_{\mathrm{LSI}}=\alpha C LSI = α R > 0 R>0 R > 0
ϱ 0 = N ( R / α e 1 , α − 1 I d ) ⊗ μ v . \varrho_{0}=N(\sqrt{R/\alpha}\,e_{1},\alpha^{-1}I_{d})\otimes\mu_{v}. ϱ 0 = N ( R / α e 1 , α − 1 I d ) ⊗ μ v . Then R 2 ( ϱ 0 ∣ μ ) = R R_{2}(\varrho_{0}\mid\mu)=R R 2 ( ϱ 0 ∣ μ ) = R ϱ t \varrho_{t} ϱ t γ > 0 \gamma>0 γ > 0 t ≥ 0 t\ge0 t ≥ 0
TV ( ϱ t , μ ) ≥ 1 2 ( 1 − e − R / 8 ) − 2 2 α π R t . (74) \operatorname{TV}(\varrho_{t},\mu)\ge\frac{1}{2}\left(1-e^{-R/8}\right)-2\sqrt{\frac{2\alpha}{\pi R}}\,t. \tag*{(74)} TV ( ϱ t , μ ) ≥ 2 1 ( 1 − e − R /8 ) − 2 π R 2 α t . (74) In particular, if R ≥ 16 R\ge16 R ≥ 16 TV ( ϱ t , μ ) > 1 / 4 \operatorname{TV}(\varrho_{t},\mu)>1/4 TV ( ϱ t , μ ) > 1/4 0 ≤ t ≤ 1 10 R / α 0\le t\le\frac{1}{10}\sqrt{R/\alpha} 0 ≤ t ≤ 10 1 R / α
Proof. The position marginals of ϱ 0 \varrho_{0} ϱ 0 μ \mu μ α − 1 I d \alpha^{-1}I_{d} α − 1 I d R / α e 1 \sqrt{R/\alpha}\,e_{1} R / α e 1 0 0 0 Σ \Sigma Σ m m m 0 0 0 m T Σ − 1 m m^{\mathsf{T}}\Sigma^{-1}m m T Σ − 1 m R 2 ( ϱ 0 ∣ μ ) = α ⋅ R / α = R R_{2}(\varrho_{0}\mid\mu)=\alpha\cdot R/\alpha=R R 2 ( ϱ 0 ∣ μ ) = α ⋅ R / α = R
Let ( X t , V t ) (X_{t},V_{t}) ( X t , V t ) ϱ 0 \varrho_{0} ϱ 0 V V V μ v \mu_{v} μ v ∣ V s , 1 ∣ |V_{s,1}| ∣ V s , 1 ∣ ∣ V 0 , 1 ∣ |V_{0,1}| ∣ V 0 , 1 ∣ [ 0 , s ] [0,s] [ 0 , s ] ∣ Z ∣ |Z| ∣ Z ∣ Z ∼ N ( 0 , 1 ) Z\sim N(0,1) Z ∼ N ( 0 , 1 ) E ∣ V s , 1 ∣ = 2 / π \mathbb{E}|V_{s,1}|=\sqrt{2/\pi} E ∣ V s , 1 ∣ = 2/ π s ≥ 0 s\ge0 s ≥ 0 Y t = α X t , 1 Y_{t}=\sqrt{\alpha}X_{t,1} Y t = α X t , 1 D t = α ∫ 0 t ∣ V s , 1 ∣ d s D_{t}=\sqrt{\alpha}\int_{0}^{t}|V_{s,1}|\,\mathrm{d}s D t = α ∫ 0 t ∣ V s , 1 ∣ d s Y t ≥ Y 0 − D t Y_{t}\ge Y_{0}-D_{t} Y t ≥ Y 0 − D t E D t = 2 α / π t \mathbb{E}D_{t}=\sqrt{2\alpha/\pi}\,t E D t = 2 α / π t
Consider the event A = { x ∈ R d : α x 1 ≥ R / 2 } A=\{x\in\mathbb{R}^{d}:\sqrt{\alpha}x_{1}\ge\sqrt{R}/2\} A = { x ∈ R d : α x 1 ≥ R /2 } μ \mu μ α x 1 ∼ N ( 0 , 1 ) \sqrt{\alpha}x_{1}\sim N(0,1) α x 1 ∼ N ( 0 , 1 ) μ ( A × R d ) ≤ 1 2 e − R / 8 \mu(A\times\mathbb{R}^{d})\le\frac{1}{2}e^{-R/8} μ ( A × R d ) ≤ 2 1 e − R /8 ϱ t \varrho_{t} ϱ t Y 0 ∼ N ( R , 1 ) Y_{0}\sim N(\sqrt{R},1) Y 0 ∼ N ( R , 1 ) R \sqrt{R} R
ϱ t ( A × R d ) ≥ P ( Y 0 ≥ R , D t ≤ R / 2 ) ≥ 1 2 − 2 E D t R = 1 2 − 2 2 α π R t . (75) \varrho_{t}(A\times\mathbb{R}^{d})\ge\mathbb{P}\left(Y_{0}\ge\sqrt{R},D_{t}\le\sqrt{R}/2\right)\ge\frac{1}{2}-\frac{2\mathbb{E}D_{t}}{\sqrt{R}}=\frac{1}{2}-2\sqrt{\frac{2\alpha}{\pi R}}\,t. \tag*{(75)} ϱ t ( A × R d ) ≥ P ( Y 0 ≥ R , D t ≤ R /2 ) ≥ 2 1 − R 2 E D t = 2 1 − 2 π R 2 α t . (75) The difference of the two probabilities of A × R d A \times\mathbb{R}^{d} A × R d R ≥ 16 R \ge16 R ≥ 16 t ≤ 1 10 R / α t \le\frac{1}{10}\sqrt{R/\alpha} t ≤ 10 1 R / α 1 2 ( 1 − e − 2 ) − 1 5 2 / π \frac{1}{2}(1-e^{-2})-\frac{1}{5}\sqrt{2/\pi} 2 1 ( 1 − e − 2 ) − 5 1 2/ π 1 / 4 1/4 1/4
For this target, Corollary 1.2 with β = C L S I = α \beta=C_{\mathrm{LSI}}=\alpha β = C LSI = α T 1 / 4 T V ≤ C α − 1 / 2 R log R T^{\mathrm{TV}}_{1/4} \le C\alpha^{-1/2}\sqrt{R\log R} T 1/4 TV ≤ C α − 1/2 R log R C C C 1 10 R / α \frac{1}{10}\sqrt{R/\alpha} 10 1 R / α log R \sqrt{\log R} log R Theorem 1.3 shows that in Rényi-2 divergence this factor cannot be removed. In dimension one, the BPS and ZZ generators coincide, so Theorem B.1 also applies to BPS when d = 1 d=1 d = 1 ∣ V ∣ \lvert V\rvert ∣ V ∣ μ v \mu_v μ v d \sqrt{d} d R / ( α d ) \sqrt{R/(\alpha d)} R / ( α d )
Theorem B.2 (Dependence on the upper Hessian bound). For κ ≥ 1 \kappa\ge1 κ ≥ 1
s = x 1 + x 2 2 , r = x 1 − x 2 2 , s=\frac{x_1+x_2}{\sqrt{2}}, \qquad r=\frac{x_1-x_2}{\sqrt{2}}, s = 2 x 1 + x 2 , r = 2 x 1 − x 2 , and let
U κ ( x ) = s 2 2 + κ r 2 2 , μ κ ( d x , d v ) ∝ e − U κ ( x ) d x μ v ( d v ) , μ v = N ( 0 , I 2 ) . (76) U_\kappa(x)=\frac{s^2}{2}+\frac{\kappa r^2}{2}, \qquad\mu_\kappa(dx,dv)\propto e^{-U_\kappa(x)}\,dx\,\mu_v(dv), \qquad\mu_v=N(0,I_2). \tag*{(76)} U κ ( x ) = 2 s 2 + 2 κ r 2 , μ κ ( d x , d v ) ∝ e − U κ ( x ) d x μ v ( d v ) , μ v = N ( 0 , I 2 ) . (76) These potentials satisfy I 2 ⪯ ∇ 2 U κ ⪯ κ I 2 I_2\preceq\nabla^2U_\kappa\preceq\kappa I_2 I 2 ⪯ ∇ 2 U κ ⪯ κ I 2 μ κ \mu_\kappa μ κ (2) with C L S I = 1 C_{\mathrm{LSI}}=1 C LSI = 1 ϱ t κ , γ \varrho_t^{\kappa,\gamma} ϱ t κ , γ U κ U_\kappa U κ γ > 0 \gamma>0 γ > 0
f 0 = d ϱ 0 κ , γ d μ κ − 1 = 1 2 sin s , f_0=\frac{d\varrho_0^{\kappa,\gamma}}{d\mu_\kappa}-1=\frac{1}{2}\sin s, f 0 = d μ κ d ϱ 0 κ , γ − 1 = 2 1 sin s , and write f t = d ϱ t κ , γ / d μ κ − 1 f_t=d\varrho_t^{\kappa,\gamma}/d\mu_\kappa-1 f t = d ϱ t κ , γ / d μ κ − 1
R ∗ = log ( 1 + 1 − e − 2 8 ) > 0 , θ ∗ = 1 4 E ∣ sin Z ∣ > 1 10 , R_*=\log\left(1+\frac{1-e^{-2}}{8}\right)>0, \qquad\theta_*=\frac{1}{4}\mathbb{E}\lvert\sin Z\rvert>\frac{1}{10}, R ∗ = log ( 1 + 8 1 − e − 2 ) > 0 , θ ∗ = 4 1 E ∣ sin Z ∣ > 10 1 , where Z ∼ N ( 0 , 1 ) Z\sim N(0,1) Z ∼ N ( 0 , 1 ) γ > 0 \gamma>0 γ > 0 t ≥ 0 t\ge0 t ≥ 0
∥ f t − f 0 ∥ L 2 ( μ κ ) 2 ≤ t κ [ 2 + log ( 1 + t κ ) ] . (77) \lVert f_t-f_0\rVert_{L^2(\mu_\kappa)}^2\le\frac{t}{\sqrt{\kappa}}\left[2+\log(1+t\sqrt{\kappa})\right]. \tag*{(77)} ∥ f t − f 0 ∥ L 2 ( μ κ ) 2 ≤ κ t [ 2 + log ( 1 + t κ ) ] . (77) Consequently, for every γ > 0 \gamma>0 γ > 0
TV ( ϱ t κ , γ , μ κ ) ≥ θ ∗ 2 and R 2 ( ϱ t κ , γ ∣ μ κ ) ≥ R ∗ 2 for 0 ≤ t ≤ κ 500 log ( e κ ) . (78) \operatorname{TV}(\varrho_t^{\kappa,\gamma},\mu_\kappa)\ge\frac{\theta_*}{2}\qquad\text{and}\qquad R_2(\varrho_t^{\kappa,\gamma}\mid\mu_\kappa)\ge\frac{R_*}{2}\qquad\text{for }0\le t\le\frac{\sqrt{\kappa}}{500\log(e\kappa)}. \tag*{(78)} TV ( ϱ t κ , γ , μ κ ) ≥ 2 θ ∗ and R 2 ( ϱ t κ , γ ∣ μ κ ) ≥ 2 R ∗ for 0 ≤ t ≤ 500 log ( e κ ) κ . (78) The time scale in (B.4) tends to infinity as κ → ∞ \kappa\to\infty κ → ∞ κ ≥ 1 \kappa\ge1 κ ≥ 1 d d d C L S I C_{\mathrm{LSI}} C LSI
Proof. The Hessian eigenvalues in the ( s , r ) (s,r) ( s , r ) 1 1 1 κ \kappa κ C L S I = 1 C_{\mathrm{LSI}}=1 C LSI = 1 [4 ] . Under μ κ \mu_\kappa μ κ s s s κ r \sqrt{\kappa}r κ r ∥ f 0 ∥ 2 = ( 1 − e − 2 ) / 8 \lVert f_0\rVert^2=(1-e^{-2})/8 ∥ f 0 ∥ 2 = ( 1 − e − 2 ) /8 R ∗ R_* R ∗ θ ∗ \theta_* θ ∗ θ ∗ > 1 / 10 \theta_*>1/10 θ ∗ > 1/10 E ∣ sin Z ∣ ≥ E sin 2 Z = ( 1 − e − 2 ) / 2 \mathbb{E}\lvert\sin Z\rvert\ge\mathbb{E}\sin^2 Z=(1-e^{-2})/2 E ∣ sin Z ∣ ≥ E sin 2 Z = ( 1 − e − 2 ) /2
Fix κ ≥ 1 \kappa\ge1 κ ≥ 1 γ > 0 \gamma> 0 γ > 0 P t = e t L \mathcal{P}_t = e^{t\mathcal{L}} P t = e t L L 2 ( μ κ ) L^2(\mu_\kappa) L 2 ( μ κ ) S = ( L + L ∗ ) / 2 \mathcal{S} = (\mathcal{L}+\mathcal{L}^*)/2 S = ( L + L ∗ ) /2
S h = 1 2 ∑ i = 1 2 ∣ v i ∂ x i U κ ∣ [ h ( x , F i v ) − h ( x , v ) ] + γ ( Π v h − h ) . (79) \mathcal{S}h=\frac{1}{2}\sum_{i=1}^{2}|v_i\partial_{x_i}U_\kappa|\left[h(x,F_i v)-h(x,v)\right]+\gamma(\Pi_vh-h). \tag*{(79)} S h = 2 1 i = 1 ∑ 2 ∣ v i ∂ x i U κ ∣ [ h ( x , F i v ) − h ( x , v ) ] + γ ( Π v h − h ) . (79) Its nonnegative quadratic form is
E ( h ) = 1 4 ∑ i = 1 2 ∫ ∣ v i ∂ x i U κ ∣ [ h ( x , F i v ) − h ( x , v ) ] 2 d μ κ + γ ∥ ( I − Π v ) h ∥ 2 . (80) \mathcal{E}(h)=\frac{1}{4}\sum_{i=1}^{2}\int|v_i\partial_{x_i}U_\kappa|\left[h(x,F_i v)-h(x,v)\right]^2\,\mathrm{d}\mu_\kappa+\gamma\|(I-\Pi_v)h\|^2. \tag*{(80)} E ( h ) = 4 1 i = 1 ∑ 2 ∫ ∣ v i ∂ x i U κ ∣ [ h ( x , F i v ) − h ( x , v ) ] 2 d μ κ + γ ∥ ( I − Π v ) h ∥ 2 . (80) Write E ( h , k ) \mathcal{E}(h,k) E ( h , k ) E ( h ) = − ⟨ h , L h ⟩ \mathcal{E}(h)=-\langle h,\mathcal{L}h\rangle E ( h ) = − ⟨ h , L h ⟩ h h h v ↦ F i v v\mapsto F_i v v ↦ F i v μ κ \mu_\kappa μ κ v i ∂ x i U κ v_i\partial_{x_i}U_\kappa v i ∂ x i U κ h ∈ D ( L ) h\in D(\mathcal{L}) h ∈ D ( L )
∫ 0 t E ( P τ h ) d τ = 1 2 ( ∥ h ∥ 2 − ∥ P t h ∥ 2 ) ≤ 1 2 ∥ h ∥ 2 , h ∈ D ( L ) . (81) \int_{0}^{t}\mathcal{E}(\mathcal{P}_\tau h)\,\mathrm{d}\tau=\frac{1}{2}\left(\|h\|^2-\|\mathcal{P}_t h\|^2\right)\le\frac{1}{2}\|h\|^2,\qquad h\in D(\mathcal{L}). \tag*{(81)} ∫ 0 t E ( P τ h ) d τ = 2 1 ( ∥ h ∥ 2 − ∥ P t h ∥ 2 ) ≤ 2 1 ∥ h ∥ 2 , h ∈ D ( L ) . (81) A parametrized resolvent. The velocity observable ( v 1 + v 2 ) / 2 (v_1+v_2)/\sqrt{2} ( v 1 + v 2 ) / 2 S \mathcal{S} S a > 0 a>0 a > 0
ξ a ( x , v ) = 1 2 ∑ i = 1 2 v i a + γ + ∣ v i ∂ x i U κ ( x ) ∣ . (82) \xi_a(x,v)=\frac{1}{\sqrt{2}}\sum_{i=1}^{2}\frac{v_i}{a+\gamma+|v_i\partial_{x_i}U_\kappa(x)|}. \tag*{(82)} ξ a ( x , v ) = 2 1 i = 1 ∑ 2 a + γ + ∣ v i ∂ x i U κ ( x ) ∣ v i . (82) Each summand is odd in its own velocity coordinate and independent of the other. Thus Π v ξ a = 0 \Pi_v\xi_a=0 Π v ξ a = 0 F i F_i F i i i i
( a − S ) ξ a = v 1 + v 2 2 . (83) (a-\mathcal{S})\xi_a=\frac{v_1+v_2}{\sqrt{2}}. \tag*{(83)} ( a − S ) ξ a = 2 v 1 + v 2 . (83) Set
b κ ( a ) = 1 2 ∑ i = 1 2 ∫ v i 2 a + ∣ v i ∂ x i U κ ( x ) ∣ d μ κ . (84) b_\kappa(a)=\frac{1}{2}\sum_{i=1}^{2}\int\frac{v_i^2}{a+|v_i\partial_{x_i}U_\kappa(x)|}\,\mathrm{d}\mu_\kappa. \tag*{(84)} b κ ( a ) = 2 1 i = 1 ∑ 2 ∫ a + ∣ v i ∂ x i U κ ( x ) ∣ v i 2 d μ κ . (84) Parity eliminates mixed velocity terms in the norm and in the form, and direct substitution gives
a ∥ ξ a ∥ 2 + E ( ξ a ) = 1 2 ∑ i = 1 2 ∫ v i 2 a + γ + ∣ v i ∂ x i U κ ( x ) ∣ d μ κ ≤ b κ ( a ) . (85) a\|\xi_a\|^2+\mathcal{E}(\xi_a)=\frac{1}{2}\sum_{i=1}^{2}\int\frac{v_i^2}{a+\gamma+|v_i\partial_{x_i}U_\kappa(x)|}\,\mathrm{d}\mu_\kappa\le b_\kappa(a). \tag*{(85)} a ∥ ξ a ∥ 2 + E ( ξ a ) = 2 1 i = 1 ∑ 2 ∫ a + γ + ∣ v i ∂ x i U κ ( x ) ∣ v i 2 d μ κ ≤ b κ ( a ) . (85) Almost invariance of f 0 f_0 f 0 As a function of x x x f 0 f_0 f 0 D ( L ) ∩ D ( L ∗ ) D(\mathcal{L})\cap D(\mathcal{L}^*) D ( L ) ∩ D ( L ∗ ) L f 0 = v ⋅ ∇ x f 0 = − L ∗ f 0 \mathcal{L}f_0=v\cdot\nabla_xf_0=-\mathcal{L}^*f_0 L f 0 = v ⋅ ∇ x f 0 = − L ∗ f 0 ω a = 1 2 cos ( s ) ξ a \omega_a=\frac{1}{2}\cos(s)\xi_a ω a = 2 1 cos ( s ) ξ a S \mathcal{S} S cos 2 s ≤ 1 \cos^2s\le1 cos 2 s ≤ 1
( a − S ) ω a = cos s 2 v 1 + v 2 2 = L f 0 = − L ∗ f 0 , a ∥ ω a ∥ 2 + E ( ω a ) ≤ b κ ( a ) 4 . (a-\mathcal{S})\omega_a=\frac{\cos s}{2}\frac{v_1+v_2}{\sqrt{2}}=\mathcal{L}f_0=-\mathcal{L}^*f_0,\qquad a\|\omega_a\|^2+\mathcal{E}(\omega_a)\le\frac{b_\kappa(a)}{4}. ( a − S ) ω a = 2 cos s 2 v 1 + v 2 = L f 0 = − L ∗ f 0 , a ∥ ω a ∥ 2 + E ( ω a ) ≤ 4 b κ ( a ) . For test functions h h h
⟨ ( a − S ) ω a , h ⟩ = a ⟨ ω a , h ⟩ + E ( ω a , h ) . \langle(a-\mathcal{S})\omega_a,h\rangle=a\langle\omega_a,h\rangle+\mathcal{E}(\omega_a,h). ⟨( a − S ) ω a , h ⟩ = a ⟨ ω a , h ⟩ + E ( ω a , h ) . Graph-norm approximation extends this identity to h ∈ D ( L ) h\in D(\mathcal{L}) h ∈ D ( L ) ω a \omega_a ω a ω a \omega_a ω a D ( L ) D(\mathcal{L}) D ( L )
Duhamel’s formula and (B.7) imply, for h ∈ D ( L ) h \in D(\mathcal{L}) h ∈ D ( L )
∣ ⟨ f t − f 0 , h ⟩ ∣ = ∣ ∫ 0 t [ a ⟨ ω a , P τ h ⟩ + E ( ω a , P τ h ) ] d τ ∣ ≤ ∥ h ∥ [ a t ∥ ω a ∥ + t 2 E ( ω a ) ] . \begin{aligned}
\left|\langle f_t-f_0,h\rangle\right|=\left|\int_0^t\left[a\langle\omega_a,\mathcal{P}_\tau h\rangle+\mathcal{E}(\omega_a,\mathcal{P}_\tau h)\right]\mathrm{d}\tau\right| \\
&\leq\|h\|\left[at\|\omega_a\|+\sqrt{\frac{t}{2}\mathcal{E}(\omega_a)}\right].
\end{aligned} ∣ ⟨ f t − f 0 , h ⟩ ∣ = ∫ 0 t [ a ⟨ ω a , P τ h ⟩ + E ( ω a , P τ h ) ] d τ ≤ ∥ h ∥ [ a t ∥ ω a ∥ + 2 t E ( ω a ) ] . Here the first term uses L 2 L^2 L 2 D ( L ) D(\mathcal{L}) D ( L ) R 2 \mathbb{R}^2 R 2
∥ f t − f 0 ∥ 2 ≤ ( a t 2 + t 2 ) [ a ∥ ω a ∥ 2 + E ( ω a ) ] ≤ 1 4 ( a t 2 + t 2 ) b κ ( a ) . (86) \|f_t-f_0\|^2\leq\left(at^2+\frac{t}{2}\right)\left[a\|\omega_a\|^2+\mathcal{E}(\omega_a)\right]\leq\frac{1}{4}\left(at^2+\frac{t}{2}\right)b_\kappa(a). \tag*{(86)} ∥ f t − f 0 ∥ 2 ≤ ( a t 2 + 2 t ) [ a ∥ ω a ∥ 2 + E ( ω a ) ] ≤ 4 1 ( a t 2 + 2 t ) b κ ( a ) . (86) The right-hand side is independent of γ \gamma γ
Bound on b κ ( a ) b_\kappa(a) b κ ( a ) Put y = κ r y=\sqrt{\kappa}r y = κ r s s s y y y μ κ \mu_\kappa μ κ κ \kappa κ
∂ x 1 U κ = s + κ y 2 , ∂ x 2 U κ = s − κ y 2 . \partial_{x_1}U_\kappa=\frac{s+\sqrt{\kappa}y}{\sqrt{2}},\qquad\partial_{x_2}U_\kappa=\frac{s-\sqrt{\kappa}y}{\sqrt{2}}. ∂ x 1 U κ = 2 s + κ y , ∂ x 2 U κ = 2 s − κ y . Each partial derivative is a centered Gaussian with variance σ κ 2 = ( 1 + κ ) / 2 \sigma_\kappa^2=(1+\kappa)/2 σ κ 2 = ( 1 + κ ) /2 v v v
b κ ( a ) = E V 2 a + σ κ ∣ V Z ∣ , b_\kappa(a)=\mathbb{E}\frac{V^2}{a+\sigma_\kappa|VZ|}, b κ ( a ) = E a + σ κ ∣ V Z ∣ V 2 , where V V V Z Z Z c > 0 c>0 c > 0 Z Z Z ∣ Z ∣ = 1 |Z|=1 ∣ Z ∣ = 1 { ∣ Z ∣ ≤ 1 } \{|Z|\leq1\} { ∣ Z ∣ ≤ 1 } ( 2 π ) − 1 / 2 < 1 / 2 (2\pi)^{-1/2}<1/2 ( 2 π ) − 1/2 < 1/2 { ∣ Z ∣ > 1 } \{|Z|>1\} { ∣ Z ∣ > 1 } a + c ∣ Z ∣ a+c|Z| a + c ∣ Z ∣ c c c P ( ∣ Z ∣ > 1 ) < 1 \mathbb{P}(|Z|>1)<1 P ( ∣ Z ∣ > 1 ) < 1
E 1 a + c ∣ Z ∣ ≤ 1 c log ( 1 + c a ) + 1 c = 1 c [ 1 + log ( 1 + c a ) ] . \mathbb{E}\frac{1}{a+c|Z|}\leq\frac{1}{c}\log\left(1+\frac{c}{a}\right)+\frac{1}{c}=\frac{1}{c}\left[1+\log\left(1+\frac{c}{a}\right)\right]. E a + c ∣ Z ∣ 1 ≤ c 1 log ( 1 + a c ) + c 1 = c 1 [ 1 + log ( 1 + a c ) ] . We condition on V V V c = σ κ ∣ V ∣ c=\sigma_\kappa|V| c = σ κ ∣ V ∣ log ( 1 + x y ) ≤ log ( 1 + x ) + log ( 1 + y ) \log(1+xy)\leq\log(1+x)+\log(1+y) log ( 1 + x y ) ≤ log ( 1 + x ) + log ( 1 + y ) x , y ≥ 0 x,y\geq0 x , y ≥ 0 E ∣ V ∣ ≤ 1 \mathbb{E}|V|\leq1 E ∣ V ∣ ≤ 1 ∣ V ∣ log ( 1 + ∣ V ∣ ) ≤ V 2 |V|\log(1+|V|)\leq V^2 ∣ V ∣ log ( 1 + ∣ V ∣ ) ≤ V 2
b κ ( a ) ≤ 1 σ κ E [ ∣ V ∣ ( 1 + log [ 1 + σ κ ∣ V ∣ a ] ) ] ≤ 2 + log ( 1 + σ κ / a ) σ κ . (87) b_\kappa(a)\leq\frac{1}{\sigma_\kappa}\mathbb{E}\left[|V|\left(1+\log\left[1+\frac{\sigma_\kappa|V|}{a}\right]\right)\right]\leq\frac{2+\log(1+\sigma_\kappa/a)}{\sigma_\kappa}. \tag*{(87)} b κ ( a ) ≤ σ κ 1 E [ ∣ V ∣ ( 1 + log [ 1 + a σ κ ∣ V ∣ ] ) ] ≤ σ κ 2 + log ( 1 + σ κ / a ) . (87) Conclusion. The case t = 0 t=0 t = 0 t > 0 t>0 t > 0 a = 1 / t a=1/t a = 1/ t κ / 2 ≤ σ κ ≤ κ \sqrt{\kappa}/2\leq\sigma_\kappa\leq\sqrt{\kappa} κ /2 ≤ σ κ ≤ κ κ ≥ 1 \kappa\geq1 κ ≥ 1 3 2 / 8 < 1 3\sqrt{2}/8<1 3 2 /8 < 1
∥ f t − f 0 ∥ 2 ≤ 3 t κ [ 2 + log ( 1 + σ κ t ) ] ≤ t κ [ 2 + log ( 1 + t κ ) ] , \|f_t-f_0\|^2\leq\frac{3t}{\sqrt{\kappa}}\left[2+\log(1+\sigma_\kappa t)\right]\leq\frac{t}{\sqrt{\kappa}}\left[2+\log(1+t\sqrt{\kappa})\right], ∥ f t − f 0 ∥ 2 ≤ κ 3 t [ 2 + log ( 1 + σ κ t ) ] ≤ κ t [ 2 + log ( 1 + t κ ) ] , which is (B.3).
For (B.4), let 0 ≤ t ≤ κ / [ 500 log ( e κ ) ] 0\leq t\leq\sqrt{\kappa}/[500\log(e\kappa)] 0 ≤ t ≤ κ / [ 500 log ( e κ )] t κ ≤ κ t\sqrt{\kappa}\leq\kappa t κ ≤ κ log ( 1 + t κ ) ≤ log ( e κ ) \log(1+t\sqrt{\kappa})\leq\log(e\kappa) log ( 1 + t κ ) ≤ log ( e κ )
∥ f t − f 0 ∥ 2 ≤ 2 + log ( e κ ) 500 log ( e κ ) ≤ 3 500 , so ∥ f t − f 0 ∥ < 0.078. \|f_t-f_0\|^2\leq\frac{2+\log(e\kappa)}{500\log(e\kappa)}\leq\frac{3}{500},
\qquad\text{so}\qquad\|f_t-f_0\|<0.078. ∥ f t − f 0 ∥ 2 ≤ 500 log ( e κ ) 2 + log ( e κ ) ≤ 500 3 , so ∥ f t − f 0 ∥ < 0.078. Since ∥ f t − f 0 ∥ L 1 ( μ κ ) ≤ ∥ f t − f 0 ∥ \|f_t-f_0\|_{L^1(\mu_\kappa)}\leq\|f_t-f_0\| ∥ f t − f 0 ∥ L 1 ( μ κ ) ≤ ∥ f t − f 0 ∥ θ ∗ > 1 / 10 \theta_*>1/10 θ ∗ > 1/10
TV ( ϱ t κ , γ , μ κ ) ≥ θ ∗ − 1 2 ∥ f t − f 0 ∥ > θ ∗ − 0.039 > θ ∗ 2 . \operatorname{TV}(\varrho_t^{\kappa,\gamma},\mu_\kappa)\geq\theta_*-\frac{1}{2}\|f_t-f_0\|>\theta_*-0.039>\frac{\theta_*}{2}. TV ( ϱ t κ , γ , μ κ ) ≥ θ ∗ − 2 1 ∥ f t − f 0 ∥ > θ ∗ − 0.039 > 2 θ ∗ . For the Rényi-2 divergence, ∥ f 0 ∥ 2 = ( 1 − e − 2 ) / 8 > 0.108 \lVert f_0\rVert^2=(1-e^{-2})/8>0.108 ∥ f 0 ∥ 2 = ( 1 − e − 2 ) /8 > 0.108 ∥ f t ∥ ≥ ∥ f 0 ∥ − ∥ f t − f 0 ∥ > 0.328 − 0.078 = 0.25 \lVert f_t\rVert\ge\lVert f_0\rVert-\lVert f_t-f_0\rVert>0.328-0.078=0.25 ∥ f t ∥ ≥ ∥ f 0 ∥ − ∥ f t − f 0 ∥ > 0.328 − 0.078 = 0.25 R 2 ( ϱ t γ , γ ? ∣ μ κ ) = log ( 1 + ∥ f t ∥ 2 ) > log ( 1.0625 ) > 0.06 \mathrm{R}_2(\varrho_t^{\gamma,\gamma?}\mid\mu_\kappa)=\log(1+\lVert f_t\rVert^2)>\log(1.0625)>0.06 R 2 ( ϱ t γ , γ ? ∣ μ κ ) = log ( 1 + ∥ f t ∥ 2 ) > log ( 1.0625 ) > 0.06 R ∗ < log ( 1.109 ) < 0.104 R_*<\log(1.109)<0.104 R ∗ < log ( 1.109 ) < 0.104
The two examples combine into a lower bound with the structure of the ZZ entry in (13) .
Corollary B.3 (Sharpness of the Zigzag bound). Let d ≥ 2 d\ge2 d ≥ 2 0 < C L S I ≤ β < ∞ 0<C_{\mathrm{LSI}}\le\beta<\infty 0 < C LSI ≤ β < ∞ R ≥ 16 R\ge16 R ≥ 16 0 < δ ≤ 1 / 20 0<\delta\le1/20 0 < δ ≤ 1/20 U U U R d \mathbb{R}^d R d C L S I I d ⪯ ∇ 2 U ⪯ β I d C_{\mathrm{LSI}}I_d\preceq\nabla^2U\preceq\beta I_d C LSI I d ⪯ ∇ 2 U ⪯ β I d (2) holds with constant C L S I C_{\mathrm{LSI}} C LSI ϱ 0 \varrho_0 ϱ 0 R 2 ( ϱ 0 ∣ μ ) ≤ R \mathrm{R}_2(\varrho_0\mid\mu)\le R R 2 ( ϱ 0 ∣ μ ) ≤ R γ > 0 \gamma>0 γ > 0
T δ T V ≥ 1 20 R C L S I + 1 1000 β C L S I log ( e β / C L S I ) . (88) T_\delta^{\mathrm{TV}}\ge\frac{1}{20}\sqrt{\frac{R}{C_{\mathrm{LSI}}}+\frac{1}{1000}\frac{\sqrt{\beta}}{C_{\mathrm{LSI}}\log(e\beta/C_{\mathrm{LSI}})}}. \tag*{(88)} T δ TV ≥ 20 1 C LSI R + 1000 1 C LSI log ( e β / C LSI ) β . (88) Proof. We construct two examples, the first with T δ T V ≥ t 1 = 1 10 R / C L S I T_\delta^{\mathrm{TV}}\ge t_1=\frac{1}{10}\sqrt{R/C_{\mathrm{LSI}}} T δ TV ≥ t 1 = 10 1 R / C LSI T δ T V ≥ t 2 = β 500 C L S I log ( e β / C L S I ) T_\delta^{\mathrm{TV}}\ge t_2=\frac{\sqrt{\beta}}{500C_{\mathrm{LSI}}\log(e\beta/C_{\mathrm{LSI}})} T δ TV ≥ t 2 = 500 C LSI l o g ( e β / C LSI ) β t 1 ≥ t 2 t_1\ge t_2 t 1 ≥ t 2 d d d C L S I C_{\mathrm{LSI}} C LSI β \beta β R R R γ \gamma γ T δ T V ≥ max { t 1 , t 2 } ≥ ( t 1 + t 2 ) / 2 T_\delta^{\mathrm{TV}}\ge\max\{t_1,t_2\}\ge(t_1+t_2)/2 T δ TV ≥ max { t 1 , t 2 } ≥ ( t 1 + t 2 ) /2 γ \gamma γ
For the first, take U ( x ) = C L S I ∣ x ∣ 2 / 2 U(x)=C_{\mathrm{LSI}}\lvert x\rvert^2/2 U ( x ) = C LSI ∣ x ∣ 2 /2 Theorem B.1 with α = C L S I \alpha=C_{\mathrm{LSI}} α = C LSI C L S I I d ⪯ β I d C_{\mathrm{LSI}}I_d\preceq\beta I_d C LSI I d ⪯ β I d R 2 ( ϱ 0 ∣ μ ) = R \mathrm{R}_2(\varrho_0\mid\mu)=R R 2 ( ϱ 0 ∣ μ ) = R R ≥ 16 R\ge16 R ≥ 16 δ ≤ 1 / 4 \delta\le1/4 δ ≤ 1/4 Theorem B.1 gives T V ( ϱ t , μ ) > 1 / 4 ≥ δ \mathrm{TV}(\varrho_t,\mu)>1/4\ge\delta TV ( ϱ t , μ ) > 1/4 ≥ δ t ≤ 1 10 R / C L S I t\le\frac{1}{10}\sqrt{R/C_{\mathrm{LSI}}} t ≤ 10 1 R / C LSI
For the second, put κ = β / C L S I ≥ 1 \kappa=\beta/C_{\mathrm{LSI}}\ge1 κ = β / C LSI ≥ 1 c > 0 c>0 c > 0 W W W R 2 \mathbb{R}^2 R 2 W c ( x ) = W ( c x ) W_c(x)=W(\sqrt{c}x) W c ( x ) = W ( c x ) ( X t , V t ) (X_t,V_t) ( X t , V t ) W W W γ \gamma γ ( X c t / c , V c t ) (X_{\sqrt{c}t}/\sqrt{c},V_{\sqrt{c}t}) ( X c t / c , V c t ) W c W_c W c c γ \sqrt{c}\gamma c γ V c t V_{\sqrt{c}t} V c t i i i t t t c ( V c t , i ∂ i W ( X c t ) ) + = ( V c t , i ∂ i W c ( X c t / c ) ) + \sqrt{c}(V_{\sqrt{c}t,i}\partial_iW(X_{\sqrt{c}t}))_+=(V_{\sqrt{c}t,i}\partial_iW_c(X_{\sqrt{c}t}/\sqrt{c}))_+ c ( V c t , i ∂ i W ( X c t ) ) + = ( V c t , i ∂ i W c ( X c t / c ) ) + c \sqrt{c} c ( x , v ) ↦ ( x / c , v ) (x,v)\mapsto(x/\sqrt{c},v) ( x , v ) ↦ ( x / c , v ) W W W W c W_c W c W = U κ W=U_\kappa W = U κ c = C L S I c=C_{\mathrm{LSI}} c = C LSI γ ↦ C L S I γ \gamma\mapsto\sqrt{C_{\mathrm{LSI}}}\gamma γ ↦ C LSI γ ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) Theorem B.2 shows that ZZ for W C L S I W_{C_{\mathrm{LSI}}} W C LSI θ ∗ / 2 > 1 / 20 ≥ δ \theta_*/2>1/20\ge\delta θ ∗ /2 > 1/20 ≥ δ 0 ≤ t ≤ κ / [ 500 C L S I log ( e κ ) ] 0\le t\le\sqrt{\kappa}/[500\sqrt{C_{\mathrm{LSI}}}\log(e\kappa)] 0 ≤ t ≤ κ / [ 500 C LSI log ( e κ )] κ / C L S I = β / C L S I \sqrt{\kappa}/\sqrt{C_{\mathrm{LSI}}}=\sqrt{\beta}/C_{\mathrm{LSI}} κ / C LSI = β / C LSI
Now define, for x ∈ R d x\in\mathbb{R}^d x ∈ R d
U ( x ) = U κ ( C L S I ( x 1 , x 2 ) ) + C L S I 2 ∑ i = 3 d x i 2 U(x)=U_\kappa\bigl(\sqrt{C_{\mathrm{LSI}}}(x_1,x_2)\bigr)+\frac{C_{\mathrm{LSI}}}{2}\sum_{i=3}^d x_i^2 U ( x ) = U κ ( C LSI ( x 1 , x 2 ) ) + 2 C LSI i = 3 ∑ d x i 2 whose Hessian has eigenvalues C L S I C_{\mathrm{LSI}} C LSI β \beta β ϱ 0 \varrho_0 ϱ 0 ( x 1 , x 2 , v 1 , v 2 ) (x_1,x_2,v_1,v_2) ( x 1 , x 2 , v 1 , v 2 ) R 2 ( ϱ 0 ∣ μ ) = R ∗ < 1 ≤ R \mathrm{R}_2(\varrho_0\mid\mu)=R_*<1\le R R 2 ( ϱ 0 ∣ μ ) = R ∗ < 1 ≤ R i ∈ { 1 , 2 } i\in\{1,2\} i ∈ { 1 , 2 } ∂ x i U \partial_{x_i}U ∂ x i U ( x 1 , x 2 ) (x_1,x_2) ( x 1 , x 2 ) ( v 1 , v 2 ) (v_1,v_2) ( v 1 , v 2 ) N ( 0 , I 2 ) \mathcal{N}(0,I_2) N ( 0 , I 2 ) ( X t , 1 , X t , 2 , V t , 1 , V t , 2 ) (X_{t,1},X_{t,2},V_{t,1},V_{t,2}) ( X t , 1 , X t , 2 , V t , 1 , V t , 2 ) W C L S I W_{C_{\mathrm{LSI}}} W C LSI ϱ t \varrho_t ϱ t T V ( ϱ t , μ ) > δ \mathrm{TV}(\varrho_t,\mu)>\delta TV ( ϱ t , μ ) > δ 0 ≤ t ≤ β / [ 500 C L S I log ( e β / C L S I ) ] 0\le t\le\sqrt{\beta}/[500C_{\mathrm{LSI}}\log(e\beta/C_{\mathrm{LSI}})] 0 ≤ t ≤ β / [ 500 C LSI log ( e β / C LSI )]
For the parameters of Corollary B.3 , Corollary 1.2, with the refreshment rate chosen as in that corollary, gives
T δ T V ≤ 6000 L C L S I 1 + R L + β C L S I ≤ 6000 C L S I ( R L + 2 L β C L S I ) , L = log 3 R 2 δ 2 . T_{\delta}^{\mathrm{TV}} \le\frac{6000\mathcal{L}}{\sqrt{C_{\mathrm{LSI}}}}\sqrt{1+\frac{R}{\mathcal{L}}+\frac{\beta}{C_{\mathrm{LSI}}}} \le\frac{6000}{\sqrt{C_{\mathrm{LSI}}}}\left(\sqrt{R\mathcal{L}}+2\mathcal{L}\sqrt{\frac{\beta}{C_{\mathrm{LSI}}}}\right), \qquad\mathcal{L}=\log\frac{3R}{2\delta^{2}}. T δ TV ≤ C LSI 6000 L 1 + L R + C LSI β ≤ C LSI 6000 ( R L + 2 L C LSI β ) , L = log 2 δ 2 3 R . where we used L ≤ L β / C L S I \mathcal{L}\le\mathcal{L}\sqrt{\beta/C_{\mathrm{LSI}}} L ≤ L β / C LSI L log ( e β / C L S I ) \mathcal{L}\log(e\beta/C_{\mathrm{LSI}}) L log ( e β / C LSI ) d ≥ 2 d\ge2 d ≥ 2 d d d C L S I C_{\mathrm{LSI}} C LSI β \beta β R R R δ \delta δ R R R 1 / δ 1/\delta 1/ δ β / C L S I \beta/C_{\mathrm{LSI}} β / C LSI β → ∞ \beta\to\infty β → ∞
The lower bounds also hold for the position marginal, for which Corollary 1.2 gives the same upper bound by data processing. In Theorem B.1 , the event A × R d A\times\mathbb{R}^{d} A × R d Theorem B.2 , the initial density f 0 f_{0} f 0 x x x ∥ Π v f t − f 0 ∥ = ∥ Π v ( f t − f 0 ) ∥ ≤ ∥ f t − f 0 ∥ \|\Pi_{v}f_{t}-f_{0}\|=\|\Pi_{v}(f_{t}-f_{0})\|\le\|f_{t}-f_{0}\| ∥ Π v f t − f 0 ∥ = ∥ Π v ( f t − f 0 ) ∥ ≤ ∥ f t − f 0 ∥ Π v f t \Pi_{v}f_{t} Π v f t
For the family of Theorem B.2 , the β \beta β (10) gives a BPS bound uniform in κ \kappa κ C L S I = 1 C_{\mathrm{LSI}}=1 C LSI = 1 d = 2 d=2 d = 2 ∇ U κ \nabla U_{\kappa} ∇ U κ s / ∣ ∇ U κ ∣ s/|\nabla U_{\kappa}| s /∣∇ U κ ∣ κ ∣ r ∣ ≫ ∣ s ∣ \kappa|r|\gg|s| κ ∣ r ∣ ≫ ∣ s ∣ μ κ \mu_{\kappa} μ κ κ \kappa κ κ \kappa κ Theorem B.2 thus identifies a genuine difference between the two dynamics. The lower bounds of this appendix are worst-case statements over the stated classes of targets and initial laws. They do not assert failure of convergence for a fixed target, and they do not concern the Gaussian cold start specifically.
References
DEPARTMENT OF MATHEMATICS, DEPARTMENT OF PHYSICS, AND DEPARTMENT OF CHEMISTRY, DUKE UNIVERSITY Email address: jianfeng@math.duke.edu
DEPARTMENT OF MATHEMATICS, DUKE UNIVERSITY Email address: yinchen.luo@duke.edu
