Introduction
Logarithmic Sobolev inequalities that are uniform in the lattice spacing and the volume give quantitative relaxation to equilibrium for the Glauber dynamics of lattice field theories [BGL ]. Bauerschmidt and Bodineau proved a logarithmic Sobolev inequality for the massive two-dimensional sine–Gordon model below β = 6 π \beta=6\pi β = 6 π BB Theorem 1.6]. Its constant is uniform in the lattice spacing for every coupling and, at small dimensionless local coupling, also in the volume. Their proof uses a multiscale Bakry–Émery criterion [BB Theorem 1.2]. The criterion uses the Polchinski flow, which averages the interaction over the Gaussian fluctuations of the field scale by scale, and assumes at each scale a lower bound, possibly negative, on the Hessian of the resulting renormalized potential, smoothed at that scale; see also [BBD2024 Theorem 3.6].
This paper concerns the kinetic counterpart of the Glauber dynamics, the underdamped Langevin dynamics
d q t = p t d t , d p t = − ∇ U ( q t ) d t − γ p t d t + 2 γ d W t , \begin{equation}
\,\mathrm dq_t=p_t\,\mathrm dt,\qquad \,\mathrm dp_t=-\nabla U(q_t)\,\mathrm dt-\gamma p_t\,\mathrm dt+\sqrt{2\gamma}\,\mathrm dW_t, \tag*{(1.1)}
\end{equation} d q t = p t d t , d p t = − ∇ U ( q t ) d t − γ p t d t + 2 γ d W t , (1.1) with friction γ > 0 \gamma>0 γ > 0 W W W
μ ( d q d p ) ∝ e − U ( q ) − ∣ p ∣ 2 / 2 d q d p . \boldsymbol\mu(\,\mathrm dq\,\,\mathrm dp)\propto e^{-U(q)-|p|^2/2}\,\mathrm dq\,\,\mathrm dp. μ ( d q d p ) ∝ e − U ( q ) − ∣ p ∣ 2 /2 d q d p . For lattice sine–Gordon, q q q U U U (1.1) is known as the canonical stochastic quantization [RyangSaitoShigemoto1985 , Horowitz1985 ]. We seek exponential decay of the relative entropy E n t ( ϱ t ∣ μ ) \mathop{\mathrm{Ent}}(\boldsymbol\varrho_t\mid\boldsymbol\mu) Ent ( ϱ t ∣ μ ) ϱ t \boldsymbol\varrho_t ϱ t ( q t , p t ) (q_t,p_t) ( q t , p t )
Two challenges make this difficult. Friction dissipates entropy only through the momentum Fisher information (2.7) , which vanishes for densities that depend on position alone. Hypocoercivity recovers decay from the transfer of dissipation by Hamiltonian transport [Villani ], and its quantitative forms [DMS , CLW , CGMZ , Lu2026 , LiLu2026 ] assume convexity of U U U ε \varepsilon ε z ≠ 0 z\neq0 z = 0 z ε = z ε − α z_\varepsilon=z\varepsilon^{-\alpha} z ε = z ε − α (2.17) , α = β / ( 4 π ) \alpha=\beta/(4\pi) α = β / ( 4 π ) ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 − ∞ -\infty − ∞ (3.8) .
Our first main result, Theorem 3.1 , is a multiscale criterion for entropy decay that measures the interaction along the Polchinski flow. Let U ( q ) = 1 2 ⟨ q , A q ⟩ + V 0 ( q ) U(q)=\frac12\left\langle q,Aq\right\rangle+V_0(q) U ( q ) = 2 1 ⟨ q , A q ⟩ + V 0 ( q ) A ⪰ I d A\succeq\mathrm{Id} A ⪰ Id V 0 V_0 V 0 s s s V s V_s V s L 2 L^2 L 2 1 1 1
E n t ( ϱ t ∣ μ ) ≤ C e − c t E n t ( ϱ 0 ∣ μ ) , t ≥ 0 , \begin{equation}
\mathop{\mathrm{Ent}}(\boldsymbol\varrho_t\mid\boldsymbol\mu)\le C\,e^{-ct}\,\mathop{\mathrm{Ent}}(\boldsymbol\varrho_0\mid\boldsymbol\mu),\qquad t\ge0, \tag*{(1.2)}
\end{equation} Ent ( ϱ t ∣ μ ) ≤ C e − c t Ent ( ϱ 0 ∣ μ ) , t ≥ 0 , (1.2) with explicit constants c , C > 0 c,C>0 c , C > 0 γ \gamma γ L 2 L^2 L 2 A A A L 2 ( μ ) L^2(\boldsymbol\mu) L 2 ( μ ) 3.3 ). The microscopic Hessian ∇ 2 V 0 \nabla^2V_0 ∇ 2 V 0
Its proof, outlined in § 3.3 , extends the Polchinski flow to phase space so that coarse-graining commutes with the free kinetic dynamics, and builds a corrector, in the sense of hypocoercivity, from the entropy decomposition of Bauerschmidt and Bodineau [BB Section 2.2, (2.16)–(2.17)].
Our second main result, Theorem 3.5 , applies this criterion to the massive lattice sine–Gordon model with 0 < β < 4 π 0<\beta<4\pi 0 < β < 4 π (1.2) holds for all lattice spacings 0 < ε ≤ 1 0<\varepsilon\le1 0 < ε ≤ 1 L ≥ 1 L\ge1 L ≥ 1 γ \gamma γ BB Sections 3.3–3.4]. The bound (3.9) on the smoothed interaction Hessian is of the microscopic size ε − α \varepsilon^{-\alpha} ε − α s − α / 2 s^{-\alpha/2} s − α /2 s = 0 s=0 s = 0 β < 4 π \beta<4\pi β < 4 π β \beta β
For friction proportional to the mass m m m m m m § 6.2 ). Under an additional small-coupling condition, the static Poincaré and logarithmic Sobolev constants of the position law are both of order m 2 m^2 m 2 6.3 ), so this rate is of the order of their square roots. For convex potentials, this square-root acceleration is established in L 2 L^2 L 2 CLW Theorem 1 and Remark 1.2(ii)] and in entropy in terms of the logarithmic Sobolev constant [Lu2026 Theorem 2.3].
Concurrently, Barashkov, Di Gesù, and Laarne [BDL2026 ] build on the space-time Poincaré inequality argument of [CLW ] to obtain exponential L 2 L^2 L 2 β < 4 π \beta<4\pi β < 4 π 6.2 . They write the interaction as cos ( β S P D E φ ) \cos(\beta_{\mathrm{SPDE}}\varphi) cos ( β SPDE φ ) BB ], so their condition β S P D E 2 < 4 π \beta_{\mathrm{SPDE}}^2<4\pi β SPDE 2 < 4 π β < 4 π \beta<4\pi β < 4 π (2.17) . For the continuum hyperbolic sine–Gordon dynamics, invariant Gibbs dynamics are constructed in [ORSW , Zine2025 ], and exponential ergodicity on the circle is proved in [Seong2024 ]. The extension to β ≥ 4 π \beta\ge4\pi β ≥ 4 π
The authors explored proof approaches with ChatGPT 5.6 Sol; the proof strategy, which extends the Polchinski flow to phase space and builds a hypocoercive corrector scale by scale, is their own. In preparing the manuscript, the authors used AI assistants, including ChatGPT 5.6 Sol, ChatGPT 6 Astra, and Claude Opus 5.5, to draft and revise the exposition and the presentation of proofs, and to check calculations and the consistency of notation. The authors verified all mathematical statements, proofs, and citations, and take full responsibility for the content of the paper.
Preliminaries
In this section, we recall some background material: § 2.1 defines the underdamped Langevin dynamics. § 2.2 recalls the Polchinski flow of Bauerschmidt and Bodineau [BB ], which averages the interaction over the Gaussian fluctuations of the position field scale by scale. § 2.3 defines the lattice sine–Gordon model.
Throughout, t t t s s s E ϱ ψ = ∫ ψ d ϱ \mathbb E_{\boldsymbol\varrho}\psi=\int\psi\,\mathrm d\boldsymbol\varrho E ϱ ψ = ∫ ψ d ϱ N ( ⋅ , ⋅ ) \mathsf N(\cdot,\cdot) N ( ⋅ , ⋅ ) ∇ k \nabla^k ∇ k k k k ∇ 0 ψ = ψ \nabla^0\psi=\psi ∇ 0 ψ = ψ q q q ∇ p \nabla_p ∇ p ∇ ( q , p ) k \nabla_{(q,p)}^k ∇ ( q , p ) k ∣ ⋅ ∣ |\cdot| ∣ ⋅ ∣ ∥ ⋅ ∥ o p \|\cdot\|_{\mathrm{op}} ∥ ⋅ ∥ op ∥ ⋅ ∥ H S \|\cdot\|_{\mathrm{HS}} ∥ ⋅ ∥ HS osc ψ = sup ψ − inf ψ \operatorname{osc}\psi = \sup\psi - \inf\psi osc ψ = sup ψ − inf ψ ψ \psi ψ
Underdamped Langevin dynamics
Let A A A d × d d\times d d × d A ⪰ I d A\succeq\mathrm{Id} A ⪰ Id
U ( q ) = 1 2 ⟨ q , A q ⟩ + V 0 ( q ) , V 0 ∈ C b ∞ ( R d ) , \begin{equation}
U(q)=\frac12\left\langle q,Aq\right\rangle+V_0(q)\,, \quad V_0\in C_b^\infty(\mathbb R^d), \tag*{(2.1)}
\end{equation} U ( q ) = 2 1 ⟨ q , A q ⟩ + V 0 ( q ) , V 0 ∈ C b ∞ ( R d ) , (2.1) where C b ∞ C_b^\infty C b ∞ (2.17) belongs to this class, although the bounds need not be uniform in the lattice spacing and the volume; see (3.8) . The corresponding Gibbs measure on the phase space R d × R d \mathbb R^d \times \mathbb R^d R d × R d Z 0 = E N ( 0 , A − 1 ) e − V 0 Z_0=\mathbb E_{\mathsf N(0,A^{-1})}e^{-V_0} Z 0 = E N ( 0 , A − 1 ) e − V 0
μ ( d q d p ) = ν ⊗ N ( 0 , I d ) = Z 0 − 1 e − V 0 ( q ) N ( 0 , A − 1 ) ( d q ) N ( 0 , I d ) ( d p ) , \begin{equation}
\boldsymbol\mu(\,\mathrm dq\,\,\mathrm dp) = \nu\otimes\mathsf N(0,\mathrm{Id}) =Z_0^{-1}e^{-V_0(q)}\mathsf N(0,A^{-1})(\,\mathrm dq)\,\mathsf N(0,\mathrm{Id})(\,\mathrm dp)\,, \tag*{(2.2)}
\end{equation} μ ( d q d p ) = ν ⊗ N ( 0 , Id ) = Z 0 − 1 e − V 0 ( q ) N ( 0 , A − 1 ) ( d q ) N ( 0 , Id ) ( d p ) , (2.2) where the position law ν ( d q ) ∝ e − U ( q ) d q ∝ e − V 0 ( q ) N ( 0 , A − 1 ) ( d q ) \nu(\,\mathrm dq) \propto e^{- U(q)} \,\mathrm dq \propto e^{-V_0(q)}\mathsf N(0,A^{-1})(\,\mathrm dq) ν ( d q ) ∝ e − U ( q ) d q ∝ e − V 0 ( q ) N ( 0 , A − 1 ) ( d q ) N ( 0 , A − 1 ) \mathsf N(0,A^{-1}) N ( 0 , A − 1 )
We consider the underdamped Langevin equation, which admits μ \boldsymbol\mu μ
d q t = p t d t , d p t = − ∇ U ( q t ) d t − γ p t d t + 2 γ d W t , \begin{equation}
\,\mathrm dq_t=p_t\,\mathrm dt,\qquad \,\mathrm dp_t=-\nabla U(q_t)\,\mathrm dt-\gamma p_t\,\mathrm dt+\sqrt{2\gamma}\,\mathrm dW_t, \tag*{(2.3)}
\end{equation} d q t = p t d t , d p t = − ∇ U ( q t ) d t − γ p t d t + 2 γ d W t , (2.3) with friction γ > 0 \gamma>0 γ > 0 W W W R d \mathbb R^d R d P t = e t L \mathcal P_t=e^{t\mathcal L} P t = e t L
L = A + γ S , A = p ⋅ ∇ q − ∇ U ⋅ ∇ p , S = Δ p − p ⋅ ∇ p . \begin{equation}
\mathcal L=\mathcal A+\gamma\mathcal S,\qquad \mathcal A=p\cdot\nabla_q-\nabla U\cdot\nabla_p, \qquad \mathcal S=\Delta_p-p\cdot\nabla_p. \tag*{(2.4)}
\end{equation} L = A + γ S , A = p ⋅ ∇ q − ∇ U ⋅ ∇ p , S = Δ p − p ⋅ ∇ p . (2.4) We write P t ∗ = e t L ∗ \mathcal P_t^*=e^{t\mathcal L^*} P t ∗ = e t L ∗ L 2 ( μ ) L^2(\boldsymbol\mu) L 2 ( μ ) μ \boldsymbol\mu μ ϱ 0 = g 0 μ \boldsymbol\varrho_0=g_0\boldsymbol\mu ϱ 0 = g 0 μ (2.3) , the law of ( q t , p t ) (q_t,p_t) ( q t , p t )
ϱ t = Law ( q t , p t ) = g t μ , g t = P t ∗ g 0 , \begin{equation}
\boldsymbol\varrho_t=\operatorname{Law}(q_t,p_t)=g_t\boldsymbol\mu,\qquad g_t=\mathcal P_t^*g_0, \tag*{(2.5)}
\end{equation} ϱ t = Law ( q t , p t ) = g t μ , g t = P t ∗ g 0 , (2.5) that is, ∂ t g t = L ∗ g t = − A g t + γ S g t \partial_tg_t=\mathcal L^*g_t=-\mathcal A g_t+\gamma\mathcal S g_t ∂ t g t = L ∗ g t = − A g t + γ S g t P t ∗ \mathcal P_t^* P t ∗ L 1 ( μ ) L^1(\boldsymbol\mu) L 1 ( μ )
We measure the convergence of ϱ t \boldsymbol\varrho_t ϱ t μ \boldsymbol\mu μ f f f μ \boldsymbol\mu μ
E n t μ ( f ) = ∫ f log f d μ , I p ( f ) = ∫ f ∣ ∇ p log f ∣ 2 d μ \begin{equation}
\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f)=\int f\log f\,\mathrm d\boldsymbol\mu,\qquad \mathcal I_p(f)=\int f|\nabla_p\log f|^2\,\mathrm d\boldsymbol\mu \tag*{(2.6)}
\end{equation} Ent μ ( f ) = ∫ f log f d μ , I p ( f ) = ∫ f ∣ ∇ p log f ∣ 2 d μ (2.6) for its relative entropy and its momentum Fisher information, with 0 log 0 = 0 0\log0=0 0 log 0 = 0 ϱ = f μ \boldsymbol\varrho=f\boldsymbol\mu ϱ = f μ E n t ( ϱ ∣ μ ) = E n t μ ( f ) \mathop{\mathrm{Ent}}(\boldsymbol\varrho\mid\boldsymbol\mu)=\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f) Ent ( ϱ ∣ μ ) = Ent μ ( f ) E n t ( ϱ ∣ μ ) = + ∞ \mathop{\mathrm{Ent}}(\boldsymbol\varrho\mid\boldsymbol\mu)=+\infty Ent ( ϱ ∣ μ ) = + ∞ ϱ \boldsymbol\varrho ϱ μ \boldsymbol\mu μ A \mathcal A A
\begin{equation}
\frac\,\mathrm d{\,\mathrm dt}\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)
=\int\log g_t\,\mathcal L^*g_t\,\mathrm d\boldsymbol\mu
=-\gamma\mathcal I_p(g_t). \tag*{(2.7)}
\end{equation}
The differentiation is justified in Lemma A.3 . We see from (2.7) that the dissipation vanishes for every density that depends only on position.
The Polchinski flow
This subsection recalls the Polchinski flow of Bauerschmidt and Bodineau [BB , BBD2024 ], which decomposes the position law ν \nu ν s ≥ 0 s\geq0 s ≥ 0 s s s q ∼ ν q\sim\nu q ∼ ν s \sqrt s s A A A s λ ≳ 1 s\lambda\gtrsim1 s λ ≳ 1 V s V_s V s s s s V s V_s V s s s s 3.1 .
We will present the Polchinski flow in the equivalent formalism of stochastic localization [Eldan2013 , ChenEldan2025 ], through noising and observation as in [Montanari2023 Sections 1.2–1.3]; [BBD2024 Section 4.5] gives a dictionary between the two points of view. At scale s s s q q q q q q s s s BB Section 1.2] is the law of this observation, up to a linear change of variables, and the renormalized potential gives its density relative to the Gaussian reference measure. The posterior, the conditional law of q q q BBD2024 ]; it provides the conditional averaging that defines coarse graining in phase space in § 4.1 . As s s s 0 0 0 q q q
q ∈ R d q \in \mathbb R^d q ∈ R d ν \nu ν s ≥ 0 s\geq0 s ≥ 0 Q s = e − s A / 2 Q_s=e^{-sA/2} Q s = e − s A /2 A A A
y s = Q s q + ξ s , \begin{equation}
y_s=Q_sq+\xi_s, \tag*{(2.8)}
\end{equation} y s = Q s q + ξ s , (2.8) where ξ s ∼ N ( 0 , C s ) \xi_s\sim\mathsf N(0,C_s) ξ s ∼ N ( 0 , C s ) q q q y s y_s y s ν s \nu_s ν s C s C_s C s q ∼ N ( 0 , A − 1 ) q\sim\mathsf N(0,A^{-1}) q ∼ N ( 0 , A − 1 ) y s ∼ N ( 0 , A − 1 ) y_s\sim\mathsf N(0,A^{-1}) y s ∼ N ( 0 , A − 1 ) q ∼ N ( 0 , A − 1 ) q\sim\mathsf N(0,A^{-1}) q ∼ N ( 0 , A − 1 ) y s y_s y s Q s A − 1 Q s + C s Q_sA^{-1}Q_s+C_s Q s A − 1 Q s + C s C s C_s C s BB (1.2)]:
C s = ( I d − Q s 2 ) A − 1 = ∫ 0 s e − u A d u . \begin{equation}
C_s = (\mathrm{Id}-Q_s^2)A^{-1} =\int_0^s e^{-uA}\,\mathrm du. \tag*{(2.9)}
\end{equation} C s = ( Id − Q s 2 ) A − 1 = ∫ 0 s e − u A d u . (2.9) The integral form shows that C s C_s C s u ≤ s u\leq s u ≤ s A − 1 = ∫ 0 ∞ e − u A d u A^{-1}=\int_0^\infty e^{-uA}\,\,\mathrm du A − 1 = ∫ 0 ∞ e − u A d u s s s C s C_s C s \frac\,\mathrm d{\,\mathrm ds}C_s=Q_s^2 , from C 0 = 0 C_0=0 C 0 = 0 y 0 = q y_0=q y 0 = q C ∞ = A − 1 C_\infty=A^{-1} C ∞ = A − 1 s → ∞ s\to\infty s → ∞ Q s → 0 Q_s\to0 Q s → 0 q q q BB ], we call s ↦ C s s\mapsto C_s s ↦ C s A − 1 A^{-1} A − 1
The choice of C s C_s C s N ( 0 , A − 1 ) \mathsf N(0,A^{-1}) N ( 0 , A − 1 ) q ∼ N ( 0 , A − 1 ) q\sim\mathsf N(0,A^{-1}) q ∼ N ( 0 , A − 1 ) ( q , y s ) (q,y_s) ( q , y s ) N ( 0 , A − 1 ) \mathsf N(0,A^{-1}) N ( 0 , A − 1 ) A − 1 Q s A^{-1}Q_s A − 1 Q s ( y s , q ) (y_s,q) ( y s , q ) ( q , y s ) (q,y_s) ( q , y s ) N ( Q s q , C s ) ( d y ) \mathsf N(Q_sq,C_s)(\,\mathrm dy) N ( Q s q , C s ) ( d y ) (2.8) , this reads
N ( 0 , A − 1 ) ( d q ) N ( Q s q , C s ) ( d y ) = N ( Q s y , C s ) ( d q ) N ( 0 , A − 1 ) ( d y ) . \begin{equation}
\mathsf N(0,A^{-1})(\,\mathrm dq)\,\mathsf N(Q_sq,C_s)(\,\mathrm dy)
=\mathsf N(Q_sy,C_s)(\,\mathrm dq)\,\mathsf N(0,A^{-1})(\,\mathrm dy). \tag*{(2.10)}
\end{equation} N ( 0 , A − 1 ) ( d q ) N ( Q s q , C s ) ( d y ) = N ( Q s y , C s ) ( d q ) N ( 0 , A − 1 ) ( d y ) . (2.10) By (2.10) , when q ∼ N ( 0 , A − 1 ) q\sim\mathsf N(0,A^{-1}) q ∼ N ( 0 , A − 1 ) q q q y s = y y_s=y y s = y N ( Q s y , C s ) \mathsf N(Q_sy,C_s) N ( Q s y , C s ) y ∈ R d y\in\mathbb R^d y ∈ R d q = Q s y + ζ q q=Q_sy+\zeta_q q = Q s y + ζ q Q s y Q_sy Q s y ζ q = q − Q s y \zeta_q=q-Q_sy ζ q = q − Q s y N ( 0 , C s ) \mathsf N(0,C_s) N ( 0 , C s ) s s s A A A q ∼ N ( 0 , A − 1 ) q\sim\mathsf N(0,A^{-1}) q ∼ N ( 0 , A − 1 ) λ − 1 \lambda^{-1} λ − 1 q q q A A A λ \lambda λ Q s y s Q_sy_s Q s y s ζ q \zeta_q ζ q
λ − 1 = λ − 1 e − s λ ⏟ coarse field + λ − 1 ( 1 − e − s λ ) ⏟ fluctuation field . \begin{equation*}
\lambda^{-1}=\underbrace{\lambda^{-1}e^{-s\lambda}}_{\text{coarse field}}+\underbrace{\lambda^{-1}(1-e^{-s\lambda})}_{\text{fluctuation field}}.
\end{equation*} λ − 1 = coarse field λ − 1 e − s λ + fluctuation field λ − 1 ( 1 − e − s λ ) . The observation thus resolves the modes with s λ ≪ 1 s\lambda\ll1 s λ ≪ 1 s λ ≫ 1 s\lambda\gg1 s λ ≫ 1 s s s
q ∼ ν q\sim\nu q ∼ ν q q q y s y_s y s π s ( d q d y ) = ν ( d q ) N ( Q s q , C s ) ( d y ) \pi_s(\,\mathrm dq\,\,\mathrm dy) = \nu(\,\mathrm dq)\,\mathsf N(Q_sq,C_s)(\,\mathrm dy) π s ( d q d y ) = ν ( d q ) N ( Q s q , C s ) ( d y ) ν ( d q ) = Z 0 − 1 e − V 0 ( q ) N ( 0 , A − 1 ) ( d q ) \nu(\,\mathrm dq)=Z_0^{-1}e^{-V_0(q)}\mathsf N(0,A^{-1})(\,\mathrm dq) ν ( d q ) = Z 0 − 1 e − V 0 ( q ) N ( 0 , A − 1 ) ( d q ) (2.10) by Z 0 − 1 e − V 0 ( q ) Z_0^{-1}e^{-V_0(q)} Z 0 − 1 e − V 0 ( q )
π s ( d q d y ) = Z 0 − 1 e − V 0 ( q ) N ( Q s y , C s ) ( d q ) N ( 0 , A − 1 ) ( d y ) . \begin{equation}
\pi_s(\,\mathrm dq\,\,\mathrm dy)=Z_0^{-1}e^{-V_0(q)}\,\mathsf N(Q_sy,C_s)(\,\mathrm dq)\,\mathsf N(0,A^{-1})(\,\mathrm dy). \tag*{(2.11)}
\end{equation} π s ( d q d y ) = Z 0 − 1 e − V 0 ( q ) N ( Q s y , C s ) ( d q ) N ( 0 , A − 1 ) ( d y ) . (2.11) By Bayes’ formula, (2.11) gives the posterior of q q q y s = y y_s=y y s = y
π s ( d q ∣ y ) ∝ e − V 0 ( q ) N ( Q s y , C s ) ( d q ) . \begin{equation*}
\pi_s(\,\mathrm dq\mid y)\propto e^{-V_0(q)}\,\mathsf N(Q_sy,C_s)(\,\mathrm dq).
\end{equation*} π s ( d q ∣ y ) ∝ e − V 0 ( q ) N ( Q s y , C s ) ( d q ) . To compute its normalizing constant, we define the renormalized potential V s ∈ C b ∞ ( R d ) V_s\in C_b^\infty(\mathbb R^d) V s ∈ C b ∞ ( R d )
e − V s ( q ) = E ζ q ∼ N ( 0 , C s ) e − V 0 ( q + ζ q ) , q ∈ R d . \begin{equation}
e^{-V_s(q)}=\mathbb E_{\zeta_q\sim\mathsf N(0,C_s)}e^{-V_0(q+\zeta_q)},\qquad q\in\mathbb R^d. \tag*{(2.12)}
\end{equation} e − V s ( q ) = E ζ q ∼ N ( 0 , C s ) e − V 0 ( q + ζ q ) , q ∈ R d . (2.12) Since q = Q s y + ζ q q=Q_sy+\zeta_q q = Q s y + ζ q (2.12) at Q s y Q_sy Q s y
∫ e − V 0 ( q ) N ( Q s y , C s ) ( d q ) = e − V s ( Q s y ) . \begin{equation}
\int e^{-V_0(q)}\,\mathsf N(Q_sy,C_s)(\,\mathrm dq) = e^{-V_s(Q_sy)}. \tag*{(2.13)}
\end{equation} ∫ e − V 0 ( q ) N ( Q s y , C s ) ( d q ) = e − V s ( Q s y ) . (2.13) We hence have
π s ( d q ∣ y ) = e V s ( Q s y ) − V 0 ( q ) N ( Q s y , C s ) ( d q ) , \begin{equation*}
\pi_s(\,\mathrm dq\mid y)=e^{V_s(Q_sy)-V_0(q)}\,\mathsf N(Q_sy,C_s)(\,\mathrm dq),
\end{equation*} π s ( d q ∣ y ) = e V s ( Q s y ) − V 0 ( q ) N ( Q s y , C s ) ( d q ) , and, for every bounded measurable function ψ \psi ψ
E π s [ ψ ( q ) ∣ y ] = e V s ( Q s y ) E ζ q ∼ N ( 0 , C s ) [ e − V 0 ( Q s y + ζ q ) ψ ( Q s y + ζ q ) ] . \begin{equation}
\mathbb E_{\pi_s}[\psi(q)\mid y]
=e^{V_s(Q_sy)}\mathbb E_{\zeta_q\sim\mathsf N(0,C_s)}\!\Big[e^{-V_0(Q_sy+\zeta_q)}\psi(Q_sy+\zeta_q)\Big]. \tag*{(2.14)}
\end{equation} E π s [ ψ ( q ) ∣ y ] = e V s ( Q s y ) E ζ q ∼ N ( 0 , C s ) [ e − V 0 ( Q s y + ζ q ) ψ ( Q s y + ζ q ) ] . (2.14) We extend this conditional expectation to phase space in § 4.1 .
(2.11) over q q q (2.13) of the posterior gives the law of the observation
ν s ( d y ) = Z 0 − 1 e − V s ( Q s y ) N ( 0 , A − 1 ) ( d y ) . \begin{equation*}
\nu_s(\,\mathrm dy)=Z_0^{-1}e^{-V_s(Q_sy)}\,\mathsf N(0,A^{-1})(\,\mathrm dy).
\end{equation*} ν s ( d y ) = Z 0 − 1 e − V s ( Q s y ) N ( 0 , A − 1 ) ( d y ) . For s < ∞ s<\infty s < ∞ y ↦ Q s y y\mapsto Q_sy y ↦ Q s y N ( 0 , A − 1 ) \mathsf N(0,A^{-1}) N ( 0 , A − 1 ) N ( 0 , A − 1 − C s ) \mathsf N(0,A^{-1}-C_s) N ( 0 , A − 1 − C s ) Q s y s Q_sy_s Q s y s
( Q s ) # ν s ( d q ) = Z 0 − 1 e − V s ( q ) N ( 0 , A − 1 − C s ) ( d q ) , \begin{equation*}
(Q_s)_\#\nu_s(\,\mathrm dq)=Z_0^{-1}e^{-V_s(q)}\,\mathsf N(0,A^{-1}-C_s)(\,\mathrm dq),
\end{equation*} ( Q s ) # ν s ( d q ) = Z 0 − 1 e − V s ( q ) N ( 0 , A − 1 − C s ) ( d q ) , which is the renormalized measure of [BB (1.5)] and [BBD2024 Definition 3.2, (3.13)] at scale s s s y y y N ( 0 , A − 1 ) \mathsf N(0,A^{-1}) N ( 0 , A − 1 ) s ∈ [ 0 , ∞ ] s\in[0,\infty] s ∈ [ 0 , ∞ ] ν s \nu_s ν s
Since ν \nu ν q q q π s \pi_s π s π s \pi_s π s y y y
ν ( d q ) = ∫ π s ( d q ∣ y ) ν s ( d y ) , \begin{equation*}
\nu(\,\mathrm dq)=\int\pi_s(\,\mathrm dq\mid y)\,\nu_s(\,\mathrm dy),
\end{equation*} ν ( d q ) = ∫ π s ( d q ∣ y ) ν s ( d y ) , the decomposition of ν \nu ν BBD2024 (3.17)–(3.19)], written in the variable y y y π s ( d q ∣ y ) \pi_s(\,\mathrm dq\mid y) π s ( d q ∣ y ) Q s y Q_sy Q s y BBD2024 ], and its pushforward under q ↦ q − Q s y q\mapsto q-Q_sy q ↦ q − Q s y ζ q \zeta_q ζ q y s = y y_s=y y s = y
s ↦ ( V s , ν s ) s\mapsto(V_s,\nu_s) s ↦ ( V s , ν s ) V 0 V_0 V 0 (2.9) . It interpolates between the position law and the Gaussian reference measure:
ν 0 = ν , ν s → N ( 0 , A − 1 ) in total variation as s → ∞ . \begin{equation*}
\nu_0=\nu,\qquad \nu_s\to\mathsf N(0,A^{-1})\ \text{in total variation as }s\to\infty.
\end{equation*} ν 0 = ν , ν s → N ( 0 , A − 1 ) in total variation as s → ∞. The first identity holds since Q 0 = I d Q_0=\mathrm{Id} Q 0 = Id C 0 = 0 C_0=0 C 0 = 0 Q s → 0 Q_s\to0 Q s → 0 C s → A − 1 C_s\to A^{-1} C s → A − 1 (2.12) .
The scale evolution of V s V_s V s (2.12) . For a matrix M M M M : ∇ 2 = ∑ i , j M i j ∂ q i ∂ q j M:\nabla^2=\sum_{i,j}M_{ij}\partial_{q_i}\partial_{q_j} M : ∇ 2 = ∑ i , j M ij ∂ q i ∂ q j (2.12) in s s s q q q \frac\,\mathrm d{\,\mathrm ds}C_s=Q_s^2 gives the heat equation
∂ s e − V s = 1 2 Q s 2 : ∇ 2 e − V s . \begin{equation*}
\partial_se^{-V_s}=\frac12Q_s^2:\nabla^2e^{-V_s}.
\end{equation*} ∂ s e − V s = 2 1 Q s 2 : ∇ 2 e − V s . By the chain rule,
Q s 2 : ∇ 2 e − V s = e − V s ( ∣ Q s ∇ V s ∣ 2 − Q s 2 : ∇ 2 V s ) , \begin{equation*}
Q_s^2:\nabla^2e^{-V_s}=e^{-V_s}\bigl(\left\lvert Q_s\nabla V_s\right\rvert^2-Q_s^2:\nabla^2V_s\bigr),
\end{equation*} Q s 2 : ∇ 2 e − V s = e − V s ( ∣ Q s ∇ V s ∣ 2 − Q s 2 : ∇ 2 V s ) , so dividing the heat equation by − e − V s -e^{-V_s} − e − V s BB , Polchinski1984 ]
∂ s V s = 1 2 Q s 2 : ∇ 2 V s − 1 2 ∣ Q s ∇ V s ∣ 2 , V s ∣ s = 0 = V 0 . \begin{equation}
\partial_sV_s=\frac12Q_s^2:\nabla^2V_s-\frac12\left\lvert Q_s\nabla V_s\right\rvert^2,\qquad V_s|_{s=0}=V_0. \tag*{(2.15)}
\end{equation} ∂ s V s = 2 1 Q s 2 : ∇ 2 V s − 2 1 ∣ Q s ∇ V s ∣ 2 , V s ∣ s = 0 = V 0 . (2.15) Its quadratic nonlinearity comes only from taking the logarithm of a solution of the linear heat equation.
The lattice sine–Gordon model
This subsection defines the massive two-dimensional lattice sine–Gordon model, to which our second main result, Theorem 3.5 , applies. We use the physical units of [BB ], in which the inner product, the Laplacian, and the noise are discretizations of their continuum counterparts. For ease of exposition, in § 3.2 and § 6.1 we pass to the unit mass when stating Theorem 3.5 and its proof.
Following [BB Section 1.3], let Ω ε , L = ε Z 2 / ( L Z 2 ) \Omega_{\varepsilon,L}=\varepsilon\mathbb Z^2/(L\mathbb Z^2) Ω ε , L = ε Z 2 / ( L Z 2 ) ε > 0 \varepsilon>0 ε > 0 L > 0 L>0 L > 0 L / ε ∈ N L/\varepsilon\in\mathbb N L / ε ∈ N L T 2 L\mathbb T^2 L T 2 φ ∈ R Ω ε , L \varphi\in\mathbb R^{\Omega_{\varepsilon,L}} φ ∈ R Ω ε , L ( L / ε ) 2 (L/\varepsilon)^2 ( L / ε ) 2 R Ω ε , L \mathbb R^{\Omega_{\varepsilon,L}} R Ω ε , L L 2 L^2 L 2
⟨ φ , φ ′ ⟩ ε = ε 2 ∑ x ∈ Ω ε , L φ x φ x ′ , ∥ φ ∥ ε 2 = ⟨ φ , φ ⟩ ε , \begin{equation*}
\left\langle \varphi,\varphi'\right\rangle_\varepsilon=\varepsilon^2\sum_{x\in\Omega_{\varepsilon,L}}\varphi_x\varphi'_x, \qquad \left\lVert \varphi\right\rVert_\varepsilon^2=\left\langle \varphi,\varphi\right\rangle_\varepsilon,
\end{equation*} ⟨ φ , φ ′ ⟩ ε = ε 2 x ∈ Ω ε , L ∑ φ x φ x ′ , ∥ φ ∥ ε 2 = ⟨ φ , φ ⟩ ε , which approximates ∫ L T 2 φ φ ′ d x \int_{L\mathbb T^2}\varphi\varphi'\,\mathrm dx ∫ L T 2 φ φ ′ d x ε → 0 \varepsilon\to0 ε → 0 R Ω ε , L \mathbb R^{\Omega_{\varepsilon,L}} R Ω ε , L ⟨ ⋅ , ⋅ ⟩ ε \left\langle \cdot,\cdot\right\rangle_\varepsilon ⟨ ⋅ , ⋅ ⟩ ε ∇ ε \nabla_\varepsilon ∇ ε ( ∇ ε ψ ) x = ε − 2 ∂ φ x ψ (\nabla_\varepsilon\psi)_x=\varepsilon^{-2}\partial_{\varphi_x}\psi ( ∇ ε ψ ) x = ε − 2 ∂ φ x ψ ψ \psi ψ R Ω ε , L \mathbb R^{\Omega_{\varepsilon,L}} R Ω ε , L
( Δ ε φ ) x : = ε − 2 ∑ j = 1 2 ( φ x + ε e j + φ x − ε e j − 2 φ x ) , \begin{equation*}
(\Delta_\varepsilon\varphi)_x := \varepsilon^{-2}\sum_{j=1}^2 \bigl(\varphi_{x+\varepsilon e_j}+\varphi_{x-\varepsilon e_j}-2\varphi_x\bigr),
\end{equation*} ( Δ ε φ ) x := ε − 2 j = 1 ∑ 2 ( φ x + ε e j + φ x − ε e j − 2 φ x ) , where e 1 , e 2 e_1,e_2 e 1 , e 2 m > 0 m>0 m > 0 β > 0 \beta>0 β > 0 z ∈ R z\in\mathbb R z ∈ R
A ε : = − Δ ε + m 2 I d , α : = β 4 π , z ε : = z ε − α . \begin{equation}
A_\varepsilon := -\Delta_\varepsilon+m^2\mathrm{Id}, \qquad \alpha := \frac{\beta}{4\pi}, \qquad z_\varepsilon := z\varepsilon^{-\alpha}. \tag*{(2.16)}
\end{equation} A ε := − Δ ε + m 2 Id , α := 4 π β , z ε := z ε − α . (2.16) The operator A ε A_\varepsilon A ε ⟨ ⋅ , ⋅ ⟩ ε \left\langle \cdot,\cdot\right\rangle_\varepsilon ⟨ ⋅ , ⋅ ⟩ ε A ε ⪰ m 2 I d A_\varepsilon\succeq m^2\mathrm{Id} A ε ⪰ m 2 Id BB ], is
H ε , L ( φ ) = 1 2 ⟨ φ , A ε φ ⟩ ε + 2 z ε ε 2 ∑ x ∈ Ω ε , L cos ( β φ x ) . \begin{equation}
H_{\varepsilon,L}(\varphi) = \frac12\left\langle \varphi,A_\varepsilon\varphi\right\rangle_\varepsilon +2z_\varepsilon\varepsilon^2 \sum_{x\in\Omega_{\varepsilon,L}}\cos\big(\sqrt\beta\,\varphi_x \big). \tag*{(2.17)}
\end{equation} H ε , L ( φ ) = 2 1 ⟨ φ , A ε φ ⟩ ε + 2 z ε ε 2 x ∈ Ω ε , L ∑ cos ( β φ x ) . (2.17) The underdamped Langevin dynamics with friction γ > 0 \gamma>0 γ > 0
d φ t = ϖ t d t , d ϖ t = − ∇ ε H ε , L ( φ t ) d t − γ ϖ t d t + 2 γ d W t ε , \begin{equation}
\,\mathrm d\varphi_t=\varpi_t\,\mathrm dt,\qquad \,\mathrm d\varpi_t=-\nabla_\varepsilon H_{\varepsilon,L}(\varphi_t)\,\mathrm dt -\gamma\varpi_t\,\mathrm dt+\sqrt{2\gamma}\,\mathrm dW_t^\varepsilon, \tag*{(2.18)}
\end{equation} d φ t = ϖ t d t , d ϖ t = − ∇ ε H ε , L ( φ t ) d t − γ ϖ t d t + 2 γ d W t ε , (2.18) where ϖ \varpi ϖ φ \varphi φ W ε W^\varepsilon W ε ( R Ω ε , L , ⟨ ⋅ , ⋅ ⟩ ε ) (\mathbb R^{\Omega_{\varepsilon,L}},\left\langle \cdot,\cdot\right\rangle_\varepsilon) ( R Ω ε , L , ⟨ ⋅ , ⋅ ⟩ ε ) W x ε = ε − 1 W x W^\varepsilon_x=\varepsilon^{-1}W_x W x ε = ε − 1 W x ( W x ) x ∈ Ω ε , L (W_x)_{x\in\Omega_{\varepsilon,L}} ( W x ) x ∈ Ω ε , L (2.18) is the canonical stochastic quantization [RyangSaitoShigemoto1985 , Horowitz1985 ] of the lattice sine–Gordon model. Its invariant measure is
μ ε , L = ν ε , L ⊗ κ ε , L , ν ε , L ( d φ ) ∝ e − H ε , L ( φ ) d φ , κ ε , L ( d ϖ ) ∝ e − ∥ ϖ ∥ ε 2 / 2 d ϖ . \begin{equation}
\boldsymbol\mu_{\varepsilon,L}=\nu_{\varepsilon,L}\otimes\kappa_{\varepsilon,L}, \quad \nu_{\varepsilon,L}(\,\mathrm d\varphi)\propto e^{-H_{\varepsilon,L}(\varphi)}\,\mathrm d\varphi, \quad \kappa_{\varepsilon,L}(\,\mathrm d\varpi)\propto e^{-\left\lVert \varpi\right\rVert_\varepsilon^2/2}\,\mathrm d\varpi. \tag*{(2.19)}
\end{equation} μ ε , L = ν ε , L ⊗ κ ε , L , ν ε , L ( d φ ) ∝ e − H ε , L ( φ ) d φ , κ ε , L ( d ϖ ) ∝ e − ∥ ϖ ∥ ε 2 /2 d ϖ . (2.19) The position marginal ν ε , L \nu_{\varepsilon,L} ν ε , L N ( 0 , A ε − 1 ) \mathsf N(0,A_\varepsilon^{-1}) N ( 0 , A ε − 1 )
ν ε , L ( d φ ) ∝ exp ( − 2 z ε ε 2 ∑ x ∈ Ω ε , L cos ( β φ x ) ) N ( 0 , A ε − 1 ) ( d φ ) . \begin{equation*}
\nu_{\varepsilon,L}(\,\mathrm d\varphi)\propto\exp\Bigl(-2z_\varepsilon\varepsilon^2\sum_{x\in\Omega_{\varepsilon,L}}\cos\big(\sqrt\beta\,\varphi_x\big)\Bigr)\mathsf N(0,A_\varepsilon^{-1})(\,\mathrm d\varphi).
\end{equation*} ν ε , L ( d φ ) ∝ exp ( − 2 z ε ε 2 x ∈ Ω ε , L ∑ cos ( β φ x ) ) N ( 0 , A ε − 1 ) ( d φ ) . The factor ε − α \varepsilon^{-\alpha} ε − α z ε z_\varepsilon z ε BB (1.16)]. For fixed m , L > 0 m,L>0 m , L > 0
V a r N ( 0 , A ε − 1 ) ( φ x ) = 1 2 π log ε − 1 + O m , L ( 1 ) , ε ↓ 0 , \begin{equation*}
\mathop{\mathrm{Var}}_{\mathsf N(0,A_\varepsilon^{-1})}(\varphi_x)
=\frac1{2\pi}\log\varepsilon^{-1}+O_{m,L}(1),\qquad \varepsilon\downarrow0,
\end{equation*} Var N ( 0 , A ε − 1 ) ( φ x ) = 2 π 1 log ε − 1 + O m , L ( 1 ) , ε ↓ 0 , so the cosine has Gaussian mean
E N ( 0 , A ε − 1 ) cos ( β φ x ) = exp ( − β 2 V a r N ( 0 , A ε − 1 ) ( φ x ) ) = ε α e O m , L ( 1 ) ⟶ 0. \begin{equation*}
\mathbb E_{\mathsf N(0,A_\varepsilon^{-1})}\cos(\sqrt\beta\,\varphi_x)
=\exp\!\left(-\frac{\beta}{2}\mathop{\mathrm{Var}}_{\mathsf N(0,A_\varepsilon^{-1})}(\varphi_x)\right)
=\varepsilon^{\alpha}e^{O_{m,L}(1)}\longrightarrow0 .
\end{equation*} E N ( 0 , A ε − 1 ) cos ( β φ x ) = exp ( − 2 β Var N ( 0 , A ε − 1 ) ( φ x ) ) = ε α e O m , L ( 1 ) ⟶ 0. Exact Wick ordering with respect to N ( 0 , A ε − 1 ) \mathsf N(0,A_\varepsilon^{-1}) N ( 0 , A ε − 1 )
: cos ( β φ x ) : = exp ( β 2 V a r N ( 0 , A ε − 1 ) ( φ x ) ) cos ( β φ x ) , \begin{equation*}
:\!\cos(\sqrt\beta\,\varphi_x)\!:\;
=\exp\!\left(\frac{\beta}{2}\mathop{\mathrm{Var}}_{\mathsf N(0,A_\varepsilon^{-1})}(\varphi_x)\right)
\cos(\sqrt\beta\,\varphi_x),
\end{equation*} : cos ( β φ x ) : = exp ( 2 β Var N ( 0 , A ε − 1 ) ( φ x ) ) cos ( β φ x ) , so that its Gaussian mean is one. The coupling z ε = z ε − α z_\varepsilon=z\varepsilon^{-\alpha} z ε = z ε − α ε − α \varepsilon^{-\alpha} ε − α (2.17) is a Wick-ordered cosine interaction whose coupling is z z z e O m , L ( 1 ) e^{O_{m,L}(1)} e O m , L ( 1 ) ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 m m m L L L α \alpha α : e i β φ : :\!e^{i\sqrt\beta\varphi}\!: : e i β φ : ν ε , L \nu_{\varepsilon,L} ν ε , L 0 < β < 8 π 0<\beta<8\pi 0 < β < 8 π z z z BB Section 1.3] for precise statements and references.
We denote the semigroup of (2.18) acting on observables by P t ε , L \mathcal P_t^{\varepsilon,L} P t ε , L L 2 ( μ ε , L ) L^2(\boldsymbol\mu_{\varepsilon,L}) L 2 ( μ ε , L ) ( P t ε , L ) ∗ (\mathcal P_t^{\varepsilon,L})^* ( P t ε , L ) ∗ g t = ( P t ε , L ) ∗ g 0 g_t=(\mathcal P_t^{\varepsilon,L})^*g_0 g t = ( P t ε , L ) ∗ g 0 ϱ t = g t μ ε , L \boldsymbol\varrho_t=g_t\boldsymbol\mu_{\varepsilon,L} ϱ t = g t μ ε , L ( φ t , ϖ t ) (\varphi_t,\varpi_t) ( φ t , ϖ t ) ε , L \varepsilon,L ε , L f f f μ ε , L \boldsymbol\mu_{\varepsilon,L} μ ε , L § 2.1 reads
E n t ( f μ ε , L ∣ μ ε , L ) = E n t μ ε , L ( f ) = ∫ f log f d μ ε , L , E μ ε , L f = 1. \begin{equation}
\mathop{\mathrm{Ent}}(f\boldsymbol\mu_{\varepsilon,L}\mid\boldsymbol\mu_{\varepsilon,L})=\mathop{\mathrm{Ent}}_{\boldsymbol\mu_{\varepsilon,L}}(f)=\int f\log f\,\mathrm d\boldsymbol\mu_{\varepsilon,L},\qquad \mathbb E_{\boldsymbol\mu_{\varepsilon,L}}f=1. \tag*{(2.20)}
\end{equation} Ent ( f μ ε , L ∣ μ ε , L ) = Ent μ ε , L ( f ) = ∫ f log f d μ ε , L , E μ ε , L f = 1. (2.20) Main results
Our main results give exponential convergence to equilibrium in relative entropy. The first is a criterion for underdamped Langevin dynamics, formulated in terms of the interaction along the Polchinski flow (§ 3.1 ). The second applies this criterion to lattice sine–Gordon and gives a decay rate uniform in the lattice spacing and the volume (§ 3.2 ). We then explain the mechanism of the proof in § 3.3 .
Entropy decay under a smoothed Hessian bound
In the coarse variable y y y V s ( Q s y ) V_s(Q_sy) V s ( Q s y )
∇ y 2 [ V s ( Q s y ) ] = Q s ∇ 2 V s ( Q s y ) Q s . \begin{equation*}
\nabla_y^2[V_s(Q_sy)]=Q_s\nabla^2V_s(Q_sy)Q_s.
\end{equation*} ∇ y 2 [ V s ( Q s y )] = Q s ∇ 2 V s ( Q s y ) Q s . We denote its uniform operator-norm bound by
b ( s ) = sup y ∈ R d ∥ Q s ∇ 2 V s ( Q s y ) Q s ∥ o p = sup q ∈ R d ∥ Q s ∇ 2 V s ( q ) Q s ∥ o p , \begin{equation*}
b(s)=\sup_{y\in\mathbb R^d}\left\lVert Q_s\nabla^2V_s(Q_sy)Q_s\right\rVert_\mathrm{op}=\sup_{q\in\mathbb R^d}\left\lVert Q_s\nabla^2V_s(q)Q_s\right\rVert_\mathrm{op},
\end{equation*} b ( s ) = y ∈ R d sup Q s ∇ 2 V s ( Q s y ) Q s op = q ∈ R d sup Q s ∇ 2 V s ( q ) Q s op , where the two suprema agree because Q s Q_s Q s
η = ( ∫ 0 ∞ b ( s ) 2 d s ) 1 / 2 . \begin{equation}
\eta=\left(\int_0^\infty b(s)^2\,\mathrm ds\right)^{1/2}. \tag*{(3.1)}
\end{equation} η = ( ∫ 0 ∞ b ( s ) 2 d s ) 1/2 . (3.1) At s = 0 s=0 s = 0 b ( 0 ) = sup q ∥ ∇ 2 V 0 ( q ) ∥ o p b(0)=\sup_q\left\lVert \nabla^2V_0(q)\right\rVert_\mathrm{op} b ( 0 ) = sup q ∇ 2 V 0 ( q ) op s > 0 s>0 s > 0 Q s Q_s Q s A A A λ \lambda λ e − s λ / 2 e^{-s\lambda/2} e − s λ /2 s λ ≫ 1 s\lambda\gg1 s λ ≫ 1 (2.12) has already averaged into V s V_s V s b ( s ) b(s) b ( s ) η \eta η L 2 L^2 L 2 b b b
The first main result of the paper is the following entropy decay estimate.
Theorem 3.1 (Entropy decay under a smoothed Hessian bound). Let A ⪰ I d A\succeq\mathrm{Id} A ⪰ Id V 0 ∈ C b ∞ ( R d ) V_0\in C_b^\infty(\mathbb R^d) V 0 ∈ C b ∞ ( R d ) γ > 0 \gamma>0 γ > 0 § 2.1 , and assume η ≤ η ∗ < 1 \eta\leq\eta_*<1 η ≤ η ∗ < 1 ϱ 0 \boldsymbol\varrho_0 ϱ 0 E n t ( ϱ 0 ∣ μ ) < ∞ \mathop{\mathrm{Ent}}(\boldsymbol\varrho_0\mid\boldsymbol\mu)<\infty Ent ( ϱ 0 ∣ μ ) < ∞ ϱ t \boldsymbol\varrho_t ϱ t t t t (2.3) with initial law ϱ 0 \boldsymbol\varrho_0 ϱ 0 E n t ( ϱ t ∣ μ ) ≤ C γ , η ∗ e − c γ , η ∗ t E n t ( ϱ 0 ∣ μ ) , t ≥ 0 ,
\mathop{\mathrm{Ent}}(\boldsymbol\varrho_t\mid\boldsymbol\mu) \leq C_{\gamma,\eta_*}\,e^{-c_{\gamma,\eta_*}t}\,\mathop{\mathrm{Ent}}(\boldsymbol\varrho_0\mid\boldsymbol\mu), \qquad t\geq0,
Ent ( ϱ t ∣ μ ) ≤ C γ , η ∗ e − c γ , η ∗ t Ent ( ϱ 0 ∣ μ ) , t ≥ 0 , c γ , η ∗ , C γ , η ∗ > 0 c_{\gamma,\eta_*},C_{\gamma,\eta_*}>0 c γ , η ∗ , C γ , η ∗ > 0 γ \gamma γ η ∗ \eta_* η ∗
The proof gives (3.2) with the explicit constants
c γ , η ∗ = ( 1 − η ∗ ) θ γ , η ∗ 1 + θ γ , η ∗ , C γ , η ∗ = 1 + θ γ , η ∗ 1 − θ γ , η ∗ , \begin{equation}
c_{\gamma,\eta_*}=\frac{(1-\eta_*)\theta_{\gamma,\eta_*}}{1+\theta_{\gamma,\eta_*}}, \qquad C_{\gamma,\eta_*}=\frac{1+\theta_{\gamma,\eta_*}}{1-\theta_{\gamma,\eta_*}}, \tag*{(3.3)}
\end{equation} c γ , η ∗ = 1 + θ γ , η ∗ ( 1 − η ∗ ) θ γ , η ∗ , C γ , η ∗ = 1 − θ γ , η ∗ 1 + θ γ , η ∗ , (3.3) where
θ γ , η ∗ = min { γ 2 K γ , η ∗ , γ ( 1 − η ∗ ) 2 ( γ + η ∗ ) 2 } , K γ , η ∗ = max { 2 , γ } + η ∗ 2 . \begin{equation}
\theta_{\gamma,\eta_*}=\min\left\{\frac{\gamma}{2K_{\gamma,\eta_*}}, \frac{\gamma(1-\eta_*)}{2(\gamma+\eta_*)^2}\right\}, \qquad K_{\gamma,\eta_*}=\max\{2,\gamma\}+\frac{\eta_*}{\sqrt2}. \tag*{(3.4)}
\end{equation} θ γ , η ∗ = min { 2 K γ , η ∗ γ , 2 ( γ + η ∗ ) 2 γ ( 1 − η ∗ ) } , K γ , η ∗ = max { 2 , γ } + 2 η ∗ . (3.4) Since 0 < θ γ , η ∗ < 1 2 0<\theta_{\gamma,\eta_*}<\frac12 0 < θ γ , η ∗ < 2 1 c γ , η ∗ > 0 c_{\gamma,\eta_*}>0 c γ , η ∗ > 0 1 < C γ , η ∗ < 3 1<C_{\gamma,\eta_*}<3 1 < C γ , η ∗ < 3 c γ , η ∗ c_{\gamma,\eta_*} c γ , η ∗ η ∗ ↑ 1 \eta_*\uparrow1 η ∗ ↑ 1 γ \gamma γ η ∗ \eta_* η ∗ d d d A ⪰ I d A\succeq\mathrm{Id} A ⪰ Id V 0 V_0 V 0 η ≤ η ∗ \eta\le\eta_* η ≤ η ∗ V 0 V_0 V 0 η \eta η
Near equilibrium, relative entropy is quadratic to leading order: E n t μ ( 1 + κ ψ ) ≈ κ 2 2 ∥ ψ ∥ 2 2 \mathop{\mathrm{Ent}}_{\boldsymbol\mu}(1+\kappa\psi)\approx\frac{\kappa^2}{2}\left\lVert \psi\right\rVert_2^2 Ent μ ( 1 + κ ψ ) ≈ 2 κ 2 ∥ ψ ∥ 2 2 κ \kappa κ ψ \psi ψ f = 1 f=1 f = 1 L 2 L^2 L 2 P t ∗ \mathcal P_t^* P t ∗ P t \mathcal P_t P t
Corollary 3.3 (L 2 L^2 L 2 3.1 , every ψ ∈ L 2 ( μ ) \psi\in L^2(\boldsymbol\mu) ψ ∈ L 2 ( μ ) E μ ψ = 0 \mathbb E_{\boldsymbol\mu}\psi=0 E μ ψ = 0 ∥ P t ψ ∥ 2 ≤ C γ , η ∗ e − c γ , η ∗ t / 2 ∥ ψ ∥ 2 .
\left\lVert \mathcal P_t\psi\right\rVert_2\le\sqrt{C_{\gamma,\eta_*}}\,e^{-c_{\gamma,\eta_*}t/2}\left\lVert \psi\right\rVert_2.
∥ P t ψ ∥ 2 ≤ C γ , η ∗ e − c γ , η ∗ t /2 ∥ ψ ∥ 2 . P t ∗ \mathcal P_t^* P t ∗
Uniform entropy decay for lattice sine–Gordon
For lattice sine–Gordon, the criterion η < 1 \eta < 1 η < 1 § 6.1 , which measures lengths and times in units of m − 1 m^{-1} m − 1 (2.18) to the same dynamics at unit mass, with the dimensionless parameters m ε m\varepsilon m ε m L mL m L z m − ( 2 − α ) z\,m^{-(2-\alpha)} z m − ( 2 − α ) γ / m \gamma/m γ / m ( ε , L , z , γ ) (\varepsilon,L,z,\gamma) ( ε , L , z , γ ) ε \varepsilon ε L L L z z z γ \gamma γ m = 1 m=1 m = 1 q = ε φ q=\varepsilon\varphi q = εφ § 6.1 then writes H ε , L H_{\varepsilon,L} H ε , L U U U (2.1) , with A = A ε = − Δ ε + I d ⪰ I d A=A_\varepsilon=-\Delta_\varepsilon+\mathrm{Id}\succeq\mathrm{Id} A = A ε = − Δ ε + Id ⪰ Id V 0 V_0 V 0 (2.17) in the variable q q q b ε , L b_{\varepsilon,L} b ε , L η ε , L \eta_{\varepsilon,L} η ε , L b b b η \eta η § 3.1 . The second main result of the paper is the following uniform entropy decay estimate.
Theorem 3.5 (Uniform entropy decay for lattice sine–Gordon). Let 0 < β < 4 π 0<\beta<4\pi 0 < β < 4 π m = 1 m=1 m = 1 z ∗ ( β ) > 0 z_*(\beta)>0 z ∗ ( β ) > 0 γ > 0 \gamma>0 γ > 0 0 < ε ≤ 1 , L ≥ 1 , L / ε ∈ N , ∣ z ∣ ≤ z ∗ ( β ) ,
0<\varepsilon\le1,\qquad L\ge1,\qquad L/\varepsilon\in\mathbb N,\qquad |z|\le z_*(\beta),
0 < ε ≤ 1 , L ≥ 1 , L / ε ∈ N , ∣ z ∣ ≤ z ∗ ( β ) , ϱ t \boldsymbol\varrho_t ϱ t (2.18) , started from any ϱ 0 \boldsymbol\varrho_0 ϱ 0 E n t ( ϱ 0 ∣ μ ε , L ) < ∞ \mathop{\mathrm{Ent}}(\boldsymbol\varrho_0\mid\boldsymbol\mu_{\varepsilon,L})<\infty Ent ( ϱ 0 ∣ μ ε , L ) < ∞ E n t ( ϱ t ∣ μ ε , L ) ≤ C γ e − c γ t E n t ( ϱ 0 ∣ μ ε , L ) , t ≥ 0 ,
\mathop{\mathrm{Ent}}(\boldsymbol\varrho_t\mid\boldsymbol\mu_{\varepsilon,L}) \le C_\gamma e^{-c_\gamma t} \mathop{\mathrm{Ent}}(\boldsymbol\varrho_0\mid\boldsymbol\mu_{\varepsilon,L}),\qquad t\ge0,
Ent ( ϱ t ∣ μ ε , L ) ≤ C γ e − c γ t Ent ( ϱ 0 ∣ μ ε , L ) , t ≥ 0 , c γ , C γ > 0 c_\gamma,C_\gamma>0 c γ , C γ > 0 γ \gamma γ
We will see in Proposition 6.1 that η ε , L ≤ 1 / 2 \eta_{\varepsilon,L}\le1/2 η ε , L ≤ 1/2 (3.5) , and hence the proof applies Theorem 3.1 with η ∗ = 1 / 2 \eta_*=1/2 η ∗ = 1/2 c γ c_\gamma c γ C γ C_\gamma C γ θ γ \theta_\gamma θ γ K γ K_\gamma K γ (3.3) and (3.4) at this value:
c γ = θ γ 2 ( 1 + θ γ ) , C γ = 1 + θ γ 1 − θ γ , \begin{equation}
c_\gamma=\frac{\theta_\gamma}{2(1+\theta_\gamma)}, \qquad C_\gamma=\frac{1+\theta_\gamma}{1-\theta_\gamma}, \tag*{(3.7)}
\end{equation} c γ = 2 ( 1 + θ γ ) θ γ , C γ = 1 − θ γ 1 + θ γ , (3.7) where
θ γ = min { γ 2 K γ , γ 4 ( γ + 1 / 2 ) 2 } , K γ = max { 2 , γ } + 1 2 2 . \begin{equation*}
\theta_\gamma=\min\left\{\frac{\gamma}{2K_\gamma},\frac{\gamma}{4(\gamma+1/2)^2}\right\}, \qquad K_\gamma=\max\{2,\gamma\}+\frac1{2\sqrt2}.
\end{equation*} θ γ = min { 2 K γ γ , 4 ( γ + 1/2 ) 2 γ } , K γ = max { 2 , γ } + 2 2 1 . As in § 3.1 , 1 < C γ < 3 1<C_\gamma<3 1 < C γ < 3 L 2 L^2 L 2 6.2 ).
The smallness assumption of Theorem 3.5 concerns the dimensionless local coupling z z z ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 L → ∞ L\to\infty L → ∞ φ x = c \varphi_x=c φ x = c x ∈ Ω ε , L x\in\Omega_{\varepsilon,L} x ∈ Ω ε , L 2 z ε ε 2 ∑ x cos ( β φ x ) 2z_\varepsilon\varepsilon^2\sum_x\cos(\sqrt\beta\,\varphi_x) 2 z ε ε 2 ∑ x cos ( β φ x ) (2.17) is a sum over the ( L / ε ) 2 (L/\varepsilon)^2 ( L / ε ) 2 Ω ε , L \Omega_{\varepsilon,L} Ω ε , L ± 2 ∣ z ∣ L 2 ε − α \pm2|z|L^2\varepsilon^{-\alpha} ± 2∣ z ∣ L 2 ε − α z cos ( β c ) = ± ∣ z ∣ z\cos(\sqrt\beta\,c)=\pm|z| z cos ( β c ) = ± ∣ z ∣ c = 0 c=0 c = 0 z ≥ 0 z\ge0 z ≥ 0 c = π / β c=\pi/\sqrt\beta c = π / β z < 0 z<0 z < 0 ∇ ε 2 H ε , L \nabla_\varepsilon^2H_{\varepsilon,L} ∇ ε 2 H ε , L ⟨ ⋅ , ⋅ ⟩ ε \left\langle \cdot,\cdot\right\rangle_\varepsilon ⟨ ⋅ , ⋅ ⟩ ε
∇ ε 2 H ε , L ( φ ) = A ε − 2 β ∣ z ∣ ε − α I d , λ min ( ∇ ε 2 H ε , L ( φ ) ) = 1 − 2 β ∣ z ∣ ε − α . \begin{equation}
\nabla_\varepsilon^2H_{\varepsilon,L}(\varphi)=A_\varepsilon-2\beta|z|\varepsilon^{-\alpha}\mathrm{Id},\qquad \lambda_{\min}\bigl(\nabla_\varepsilon^2H_{\varepsilon,L}(\varphi)\bigr)=1-2\beta|z|\varepsilon^{-\alpha}. \tag*{(3.8)}
\end{equation} ∇ ε 2 H ε , L ( φ ) = A ε − 2 β ∣ z ∣ ε − α Id , λ m i n ( ∇ ε 2 H ε , L ( φ ) ) = 1 − 2 β ∣ z ∣ ε − α . (3.8) For any fixed nonzero z z z ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 L → ∞ L\to\infty L → ∞ (3.8) tends to − ∞ -\infty − ∞ ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 ε \varepsilon ε
Theorem 3.5 is uniform in these limits because the constants of Theorem 3.1 depend on the interaction only through η \eta η b ε , L b_{\varepsilon,L} b ε , L η ε , L \eta_{\varepsilon,L} η ε , L b ε , L b_{\varepsilon,L} b ε , L 3.5 , the proof of Proposition 6.1 gives
b ε , L ( s ) ≤ C β ∣ z ∣ { ε − α , 0 ≤ s ≤ ε 2 , s − α / 2 , ε 2 ≤ s ≤ 1 , e − s , s ≥ 1 , α = β 4 π , \begin{equation}
b_{\varepsilon,L}(s)\leq C_\beta|z|
\begin{cases}
\varepsilon^{-\alpha},&0\leq s\leq\varepsilon^2,\\
s^{-\alpha/2},&\varepsilon^2\leq s\leq1,\\
e^{-s},&s\geq1,
\end{cases}
\qquad \alpha=\frac{\beta}{4\pi}, \tag*{(3.9)}
\end{equation} b ε , L ( s ) ≤ C β ∣ z ∣ ⎩ ⎨ ⎧ ε − α , s − α /2 , e − s , 0 ≤ s ≤ ε 2 , ε 2 ≤ s ≤ 1 , s ≥ 1 , α = 4 π β , (3.9) with C β C_\beta C β ε \varepsilon ε L L L s − α / 2 s^{-\alpha/2} s − α /2 s \sqrt{s} s β \beta β
∫ 0 1 s − α d s < ∞ ⟺ α < 1 ⟺ β < 4 π . \begin{equation*}
\int_0^1s^{-\alpha}\,\mathrm ds<\infty
\quad\Longleftrightarrow\quad \alpha<1
\quad\Longleftrightarrow\quad \beta<4\pi.
\end{equation*} ∫ 0 1 s − α d s < ∞ ⟺ α < 1 ⟺ β < 4 π . Under this condition, the contribution of the interval [ 0 , ε 2 ] [0,\varepsilon^2] [ 0 , ε 2 ] η ε , L 2 \eta_{\varepsilon,L}^2 η ε , L 2 z 2 ε 2 − 2 α z^2\varepsilon^{2-2\alpha} z 2 ε 2 − 2 α ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0
For general mass m > 0 m>0 m > 0 § 6.1 gives the physical-time rate c γ m c_\gamma m c γ m ε \varepsilon ε L L L z z z γ \gamma γ m m m § 6.2 .
Strategy of the proof
We outline the ideas behind Theorem 3.1 . Hamiltonian transport turns position dependence into momentum dependence, and hypocoercivity quantifies this transfer by adding to the entropy a cross term that couples position and momentum derivatives [Villani ]. The kinetic derivative of a cross term built from the density itself contains the Hessian of U U U − ∞ -\infty − ∞ ε → 0 \varepsilon \to 0 ε → 0 (3.8) . We build the cross term from the coarse-grained densities of the Polchinski flow, in which the interaction enters through the smoothed interaction Hessian, and integrate it over all scales; this is the corrector (5.1) of § 5.1 . § 4 –§ 5 carry out the argument for smooth initial densities bounded away from zero, and Appendix § 7 supplies the regularity estimates and the approximation of general initial laws of finite entropy.
Differentiating the cross term at a fixed scale requires a kinetic equation for the coarse-grained density: evolving a law and then observing it should give the same law as observing first and then evolving (Figure 1 ). The position observation is that of the Polchinski flow, and we match the momentum observation to it (§ 4.1 ). On an eigenmode of A A A (?) ).
The corrector is built on the entropy decomposition of Bauerschmidt and Bodineau, which writes entropy as the information that the observation loses about the state, summed over all scales (Lemma 4.2 ). The rate of loss is a Fisher information that involves both the position and momentum scores, the logarithmic gradients of the coarse-grained density, whereas friction dissipates entropy through momentum alone. The corrector pairs the two scores at each scale and integrates the pairing over scales. Free Hamiltonian transport rotates position dependence into momentum dependence at every scale, so the kinetic derivative of the corrector produces the position part of the entropy decomposition with a negative sign (Lemma 5.3 ). The decomposition also bounds the corrector by entropy, so the modified entropy is comparable to entropy (Lemma 5.1 ).
The remaining terms in the kinetic derivative of the corrector involve the coarse-grained densities at every scale, while friction controls only the momentum Fisher information of the density itself, at scale zero. Coarse-graining bridges this mismatch of scales. In the momentum variable, the Polchinski flow is an Ornstein–Uhlenbeck smoothing determined by A A A 5.4 ). The interaction enters only through the force defect, which vanishes at scale zero and is generated along the flow by the smoothed interaction Hessian acting on position derivatives. Since the entropy decomposition controls position derivatives in mean square over scales, the accumulated force defect is measured against entropy by the L 2 L^2 L 2 η \eta η b b b 5.5 ). It removes at most a fraction η \eta η η ≤ η ∗ < 1 \eta\leq\eta_*<1 η ≤ η ∗ < 1 5.2 ).
Coarse graining and kinetic evolution
We construct the observation laws and derive their evolution in both coarse-graining scale and kinetic time, establishing the commuting diagram in Figure 1 , which is fundamental for constructing the corrector in the modified entropy functional.
Matched coarse graining in phase space
In this subsection, we extend the position-space Polchinski flow of § 2.2 to phase space through a Gaussian observation of position and momentum. We match the momentum observation to the position observation such that the observation process commutes with the free kinetic dynamics, (2.3) with V 0 = 0 V_0=0 V 0 = 0 Figure 1 .
[>=stealth,
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minimum width=3.4cm,font=\small},
lab/.style={font=\small,align=center},
scale arrow/.style={->,semithick,draw=RoyalBlue!70!black},
kinetic arrow/.style={->,semithick,draw=RoyalRed!85!black}]
\node[lab] at (3,-0.6) {kinetic time $0$};
\node[lab] at (12.3,-0.6) {kinetic time $t$};
\node[lab,anchor=east] at (1,-1.25) {$s=0$};
\node[lab,anchor=east] at (1,-3.25) {scale $s$};
\node[lab,anchor=east] at (1,-4.4) {$s\to\infty$};
\node[state] (f0) at (3,-1.25) {$(q_0,p_0)\sim\boldsymbol\varrho_0$};
\node[state] (ft) at (12.3,-1.25) {$(q_t,p_t)\sim\boldsymbol\varrho_t$};
\node[state] (c0) at (3,-3.25) {$(y_{0,s},w_{0,s})\sim\boldsymbol\varrho_{0,s}$};
\node[state] (ct) at (12.3,-3.25) {$(y_{t,s},w_{t,s})\sim\boldsymbol\varrho_{t,s}$};
\node[lab] (e0) at (3,-4.4) {$\mathsf N(0,\mathbf C_\infty)$};
\node[lab] (et) at (12.3,-4.4) {$\mathsf N(0,\mathbf C_\infty)$};
\draw[kinetic arrow] (f0) -- node[lab,above]{underdamped Langevin \eqref{eq:langevindy}}
node[lab,below]{$g_t=\mathcal P_t^*g_0$} (ft);
\draw[kinetic arrow] (c0) -- node[lab,above]{coarse kinetic equation \eqref{eq:ent-coarse-density-equation}}
node[lab,below]{$\partial_tg_{t,s}=-\mathcal A_sg_{t,s}+\gamma\mathcal S_wg_{t,s}-\nabla_w^*R_{t,s}$} (ct);
\draw[scale arrow] (f0) -- node[lab,left]{Gaussian observation\\
$g_{0,s}=\mathsf P_sg_0$} (c0);
\draw[scale arrow] (ft) -- node[lab,left]{Gaussian observation\\
$g_{t,s}=\mathsf P_sg_t$} (ct);
\draw[scale arrow,dashed] (c0) -- (e0);
\draw[scale arrow,dashed] (ct) -- (et);Figure 1. Kinetic evolution and Gaussian observation. Horizontal (red) arrows follow kinetic time at fixed scale, and vertical (blue) arrows follow scale at fixed kinetic time, along the phase-space Polchinski flow. The state ( q t , p t ) (q_t,p_t) ( q t , p t ) ϱ t \boldsymbol\varrho_t ϱ t ( y t , s , w t , s ) (y_{t,s},w_{t,s}) ( y t , s , w t , s ) ϱ t , s \boldsymbol\varrho_{t,s} ϱ t , s g t g_t g t g t , s g_{t,s} g t , s μ \boldsymbol\mu μ μ s \boldsymbol\mu_s μ s P s \mathsf P_s P s (4.7) . The lower row is the exact evolution of g t , s g_{t,s} g t , s R t , s R_{t,s} R t , s V 0 = 0 V_0=0 V 0 = 0 s → ∞ s\to\infty s → ∞
C ∞ : = diag ( A − 1 , I d ) \mathbf C_\infty:=\operatorname{diag}(A^{-1},\mathrm{Id}) C ∞ := diag ( A − 1 , Id ) (2.2) is μ ( d q d p ) = Z 0 − 1 e − V 0 ( q ) N ( 0 , C ∞ ) ( d q d p ) \boldsymbol\mu(\,\mathrm dq\,\,\mathrm dp)=Z_0^{-1}e^{-V_0(q)}\,\mathsf N(0,\mathbf C_\infty)(\,\mathrm dq\,\,\mathrm dp) μ ( d q d p ) = Z 0 − 1 e − V 0 ( q ) N ( 0 , C ∞ ) ( d q d p ) N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ ) V 0 V_0 V 0 Q s Q_s Q s Q s = diag ( Q s , Q s ) \mathbf Q_s=\operatorname{diag}(Q_s,Q_s) Q s = diag ( Q s , Q s ) s ≥ 0 s\geq0 s ≥ 0 ( q , p ) (q,p) ( q , p )
( y s w s ) = Q s ( q p ) + ξ s , ξ s ∼ N ( 0 , C s ) , \begin{equation}
\begin{pmatrix}y_s\\w_s\end{pmatrix}
=\mathbf Q_s\begin{pmatrix}q\\p\end{pmatrix}+\boldsymbol\xi_s,
\qquad \boldsymbol\xi_s\sim\mathsf N(0,\mathbf C_s), \tag*{(4.1)}
\end{equation} ( y s w s ) = Q s ( q p ) + ξ s , ξ s ∼ N ( 0 , C s ) , (4.1) with ( q , p ) (q,p) ( q , p ) ξ s \boldsymbol\xi_s ξ s ( y s , w s ) (y_s,w_s) ( y s , w s ) ( q , p ) (q,p) ( q , p ) (2.9) , the observation process (4.1) preserves the reference measure N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ ) C ∞ = Q s C ∞ Q s + C s \mathbf C_\infty=\mathbf Q_s\mathbf C_\infty\mathbf Q_s+\mathbf C_s C ∞ = Q s C ∞ Q s + C s s s s Q s \mathbf Q_s Q s C ∞ \mathbf C_\infty C ∞
\begin{equation}
\mathbf C_s = (\mathrm{Id}-\mathbf Q_s^2)\mathbf C_\infty = \begin{pmatrix}C_s&0\\0&AC_s\end{pmatrix},\qquad
\frac\,\mathrm d{\,\mathrm ds}\mathbf C_s=\begin{pmatrix}Q_s^2&0\\0&AQ_s^2\end{pmatrix}. \tag*{(4.2)}
\end{equation}
We call this matched coarse graining, since the same fraction I d − Q s 2 \mathrm{Id}-Q_s^2 Id − Q s 2 C 0 = 0 \mathbf C_0=0 C 0 = 0 \frac\,\mathrm d{\,\mathrm ds}\mathbf C_s\succeq0 , and C s → C ∞ \mathbf C_s\to\mathbf C_\infty C s → C ∞ s → ∞ s\to\infty s → ∞ s ↦ C s s\mapsto\mathbf C_s s ↦ C s C ∞ \mathbf C_\infty C ∞ R 2 d \mathbb R^{2d} R 2 d BB Section 2.1].
V 0 = 0 V_0=0 V 0 = 0 (2.3) is the Gaussian process
d ( q t p t ) = M ( q t p t ) d t + ( 0 2 γ d W t ) , M = ( 0 I d − A − γ I d ) . \begin{equation}
\,\mathrm d\begin{pmatrix}q_t\\p_t\end{pmatrix}
=\mathbf M\begin{pmatrix}q_t\\p_t\end{pmatrix}\,\mathrm dt
+\begin{pmatrix}0\\\sqrt{2\gamma}\,\,\mathrm dW_t\end{pmatrix},
\qquad \mathbf M=\begin{pmatrix}0&\mathrm{Id}\\-A&-\gamma\mathrm{Id}\end{pmatrix}. \tag*{(4.3)}
\end{equation} d ( q t p t ) = M ( q t p t ) d t + ( 0 2 γ d W t ) , M = ( 0 − A Id − γ Id ) . (4.3) Since Q s \mathbf Q_s Q s M \mathbf M M (4.1) and the free kinetic dynamics (4.3) preserve N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ ) ( q , p ) (q,p) ( q , p )
N ( Q s e t M ( q , p ) ⊤ , C ∞ − Q s e t M C ∞ e t M ⊤ Q s ) . \begin{equation*}
\mathsf N\bigl(\mathbf Q_se^{t\mathbf M}(q,p)^\top,\ \mathbf C_\infty-\mathbf Q_se^{t\mathbf M}\mathbf C_\infty e^{t\mathbf M^\top}\mathbf Q_s\bigr).
\end{equation*} N ( Q s e t M ( q , p ) ⊤ , C ∞ − Q s e t M C ∞ e t M ⊤ Q s ) . This is the reason for the common attenuation Q s Q_s Q s diag ( Q , Q ′ ) \operatorname{diag}(Q,Q') diag ( Q , Q ′ ) M \mathbf M M Q ′ = Q Q'=Q Q ′ = Q Q Q Q A A A N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ ) (4.2) . Hence, for V 0 = 0 V_0=0 V 0 = 0 (4.3) gives the same law as evolving and then observing: the bottom row of Figure 1 is then the free kinetic dynamics, the same as the top row.
§ 2.2 , we use the reversibility of the Gaussian observation kernel to identify the observation law and the conditional expectation that defines coarse graining. The observation (4.1) takes values in the fixed coarse space R 2 d \mathbb R^{2d} R 2 d ( y , w ) (y,w) ( y , w ) Q s \mathbf Q_s Q s C ∞ \mathbf C_\infty C ∞ Q s C ∞ Q s + C s = C ∞ \mathbf Q_s\mathbf C_\infty\mathbf Q_s+\mathbf C_s=\mathbf C_\infty Q s C ∞ Q s + C s = C ∞ (2.10) gives
N ( 0 , C ∞ ) ( d q d p ) N ( Q s ( q , p ) ⊤ , C s ) ( d y d w ) = N ( Q s ( y , w ) ⊤ , C s ) ( d q d p ) N ( 0 , C ∞ ) ( d y d w ) . \begin{equation}
\mathsf N(0,\mathbf C_\infty)(\,\mathrm dq\,\,\mathrm dp)\,\mathsf N\bigl(\mathbf Q_s(q,p)^\top,\mathbf C_s\bigr)(\,\mathrm dy\,\,\mathrm dw)
=\mathsf N\bigl(\mathbf Q_s(y,w)^\top,\mathbf C_s\bigr)(\,\mathrm dq\,\,\mathrm dp)\,\mathsf N(0,\mathbf C_\infty)(\,\mathrm dy\,\,\mathrm dw). \tag*{(4.4)}
\end{equation} N ( 0 , C ∞ ) ( d q d p ) N ( Q s ( q , p ) ⊤ , C s ) ( d y d w ) = N ( Q s ( y , w ) ⊤ , C s ) ( d q d p ) N ( 0 , C ∞ ) ( d y d w ) . (4.4) The joint law π s \boldsymbol\pi_s π s ( q , p ) ∼ μ (q,p)\sim\boldsymbol\mu ( q , p ) ∼ μ ( y s , w s ) (y_s,w_s) ( y s , w s ) μ ( d q d p ) N ( Q s ( q , p ) ⊤ , C s ) ( d y d w ) \boldsymbol\mu(\,\mathrm dq\,\,\mathrm dp)\,\mathsf N\bigl(\mathbf Q_s(q,p)^\top,\mathbf C_s\bigr)(\,\mathrm dy\,\,\mathrm dw) μ ( d q d p ) N ( Q s ( q , p ) ⊤ , C s ) ( d y d w ) (2.2) and multiplying (4.4) by Z 0 − 1 e − V 0 ( q ) Z_0^{-1}e^{-V_0(q)} Z 0 − 1 e − V 0 ( q )
π s ( d q d p d y d w ) = Z 0 − 1 e − V 0 ( q ) N ( 0 , C ∞ ) ( d q d p ) N ( Q s ( q , p ) ⊤ , C s ) ( d y d w ) = Z 0 − 1 e − V 0 ( q ) N ( Q s ( y , w ) ⊤ , C s ) ( d q d p ) N ( 0 , C ∞ ) ( d y d w ) , \begin{align}
\boldsymbol\pi_s(\,\mathrm dq\,\,\mathrm dp\,\,\mathrm dy\,\,\mathrm dw)
&{}=Z_0^{-1}e^{-V_0(q)}\,\mathsf N(0,\mathbf C_\infty)(\,\mathrm dq\,\,\mathrm dp)\,\mathsf N\bigl(\mathbf Q_s(q,p)^\top,\mathbf C_s\bigr)(\,\mathrm dy\,\,\mathrm dw) \notag \\
&=Z_0^{-1}e^{-V_0(q)}\,\mathsf N\bigl(\mathbf Q_s(y,w)^\top,\mathbf C_s\bigr)(\,\mathrm dq\,\,\mathrm dp)\,\mathsf N(0,\mathbf C_\infty)(\,\mathrm dy\,\,\mathrm dw), \tag*{(4.5)}
\end{align} π s ( d q d p d y d w ) = Z 0 − 1 e − V 0 ( q ) N ( 0 , C ∞ ) ( d q d p ) N ( Q s ( q , p ) ⊤ , C s ) ( d y d w ) = Z 0 − 1 e − V 0 ( q ) N ( Q s ( y , w ) ⊤ , C s ) ( d q d p ) N ( 0 , C ∞ ) ( d y d w ) , (4.5) which extends the joint law π s \pi_s π s (2.11) to phase space. Since V 0 V_0 V 0 q q q C s \mathbf C_s C s (4.5) over ( q , p ) (q,p) ( q , p ) (2.12) gives the observation law, which we call the renormalized phase-space measure:
μ s ( d y d w ) = Z 0 − 1 e − V s ( Q s y ) N ( 0 , C ∞ ) ( d y d w ) = ν s ( d y ) N ( 0 , I d ) ( d w ) . \begin{equation}
\boldsymbol\mu_s(\,\mathrm dy\,\,\mathrm dw)=Z_0^{-1}e^{-V_s(Q_sy)}\,\mathsf N(0,\mathbf C_\infty)(\,\mathrm dy\,\,\mathrm dw)=\nu_s(\,\mathrm dy)\,\mathsf N(0,\mathrm{Id})(\,\mathrm dw). \tag*{(4.6)}
\end{equation} μ s ( d y d w ) = Z 0 − 1 e − V s ( Q s y ) N ( 0 , C ∞ ) ( d y d w ) = ν s ( d y ) N ( 0 , Id ) ( d w ) . (4.6) In the coarse variables, the renormalized potential enters through the composition V s ( Q s y ) V_s(Q_sy) V s ( Q s y ) ∇ y [ V s ( Q s y ) ] = Q s ∇ V s ( Q s y ) \nabla_y[V_s(Q_sy)]=Q_s\nabla V_s(Q_sy) ∇ y [ V s ( Q s y )] = Q s ∇ V s ( Q s y )
Bayes’ formula gives the posterior
π s ( d q d p ∣ y , w ) = e V s ( Q s y ) − V 0 ( q ) N ( Q s ( y , w ) ⊤ , C s ) ( d q d p ) . \begin{equation*}
\boldsymbol\pi_s(\,\mathrm dq\,\,\mathrm dp\mid y,w)
=e^{V_s(Q_sy)-V_0(q)}\,\mathsf N\bigl(\mathbf Q_s(y,w)^\top,\mathbf C_s\bigr)(\,\mathrm dq\,\,\mathrm dp).
\end{equation*} π s ( d q d p ∣ y , w ) = e V s ( Q s y ) − V 0 ( q ) N ( Q s ( y , w ) ⊤ , C s ) ( d q d p ) . We define the coarse-graining operator by the conditional expectation given the observation, i.e., P s ψ ( y , w ) = E π s [ ψ ( q , p ) ∣ y , w ] \mathsf P_s\psi(y,w)=\mathbb E_{\boldsymbol\pi_s}[\psi(q,p)\mid y,w] P s ψ ( y , w ) = E π s [ ψ ( q , p ) ∣ y , w ] ψ ∈ L 1 ( μ ) \psi\in L^1(\boldsymbol\mu) ψ ∈ L 1 ( μ ) ψ ≥ 0 \psi\geq0 ψ ≥ 0
P s ψ ( y , w ) = e V s ( Q s y ) E ( ζ q , ζ p ) ∼ N ( 0 , C s ) [ e − V 0 ( Q s y + ζ q ) ψ ( Q s y + ζ q , Q s w + ζ p ) ] , \begin{equation}
\mathsf P_s\psi(y,w)
=e^{V_s(Q_sy)}\mathbb E_{{(\zeta_q,\zeta_p)\sim\mathsf N(0,\mathbf C_s)}}\!\left[e^{-V_0(Q_sy+\zeta_q)}\psi(Q_sy+\zeta_q,Q_sw+\zeta_p)\right], \tag*{(4.7)}
\end{equation} P s ψ ( y , w ) = e V s ( Q s y ) E ( ζ q , ζ p ) ∼ N ( 0 , C s ) [ e − V 0 ( Q s y + ζ q ) ψ ( Q s y + ζ q , Q s w + ζ p ) ] , (4.7) where ( ζ q , ζ p ) (\zeta_q,\zeta_p) ( ζ q , ζ p ) Q s ( y , w ) ⊤ \mathbf Q_s(y,w)^\top Q s ( y , w ) ⊤ P s \mathsf P_s P s (2.14) . The pushforward ( Q s ) # μ s (\mathbf Q_s)_\#\boldsymbol\mu_s ( Q s ) # μ s ψ ↦ ( P s ψ ) ∘ Q s − 1 \psi\mapsto(\mathsf P_s\psi)\circ\mathbf Q_s^{-1} ψ ↦ ( P s ψ ) ∘ Q s − 1 § 8 for the details.
Let f f f μ \boldsymbol\mu μ f s = P s f f_s=\mathsf P_sf f s = P s f ( q , p ) (q,p) ( q , p ) f μ f\boldsymbol\mu f μ ( y s , w s ) (y_s,w_s) ( y s , w s ) f s μ s f_s\boldsymbol\mu_s f s μ s μ \boldsymbol\mu μ μ s \boldsymbol\mu_s μ s ( q , p ) (q,p) ( q , p ) ( y s , w s ) (y_s,w_s) ( y s , w s ) f ( q , p ) π s ( d q d p d y d w ) f(q,p)\,\boldsymbol\pi_s(\,\mathrm dq\,\,\mathrm dp\,\,\mathrm dy\,\,\mathrm dw) f ( q , p ) π s ( d q d p d y d w ) ( q , p ) (q,p) ( q , p ) f μ f\boldsymbol\mu f μ P s \mathsf P_s P s ( y , w ) (y,w) ( y , w ) f s μ s f_s\boldsymbol\mu_s f s μ s P s \mathsf P_s P s f s μ s f_s\boldsymbol\mu_s f s μ s
P s 1 = 1 , ∫ f s d μ s = ∫ f d μ = 1. \begin{equation}
\mathsf P_s1=1,\qquad \int f_s\,\mathrm d\boldsymbol\mu_s=\int f\,\mathrm d\boldsymbol\mu=1. \tag*{(4.8)}
\end{equation} P s 1 = 1 , ∫ f s d μ s = ∫ f d μ = 1. (4.8) At s = 0 s=0 s = 0 Q 0 = I d \mathbf Q_0=\mathrm{Id} Q 0 = Id C 0 = 0 \mathbf C_0=0 C 0 = 0 μ 0 = μ \boldsymbol\mu_0=\boldsymbol\mu μ 0 = μ f 0 = f f_0=f f 0 = f s ↦ ( μ s , f s ) s\mapsto(\boldsymbol\mu_s,f_s) s ↦ ( μ s , f s ) ( q t , p t ) ∼ ϱ t (q_t,p_t)\sim\boldsymbol\varrho_t ( q t , p t ) ∼ ϱ t ( y t , s , w t , s ) ⊤ = Q s ( q t , p t ) ⊤ + ξ s (y_{t,s},w_{t,s})^\top=\mathbf Q_s(q_t,p_t)^\top+\boldsymbol\xi_s ( y t , s , w t , s ) ⊤ = Q s ( q t , p t ) ⊤ + ξ s ξ s ∼ N ( 0 , C s ) \boldsymbol\xi_s\sim\mathsf N(0,\mathbf C_s) ξ s ∼ N ( 0 , C s ) ( y t , s , w t , s ) (y_{t,s},w_{t,s}) ( y t , s , w t , s ) μ s \boldsymbol\mu_s μ s
g t , s = P s g t , ϱ t , s ( d y d w ) = g t , s ( y , w ) μ s ( d y d w ) . \begin{equation}
g_{t,s}=\mathsf P_sg_t,\qquad
\boldsymbol\varrho_{t,s}(\,\mathrm dy\,\,\mathrm dw)=g_{t,s}(y,w)\,\boldsymbol\mu_s(\,\mathrm dy\,\,\mathrm dw). \tag*{(4.9)}
\end{equation} g t , s = P s g t , ϱ t , s ( d y d w ) = g t , s ( y , w ) μ s ( d y d w ) . (4.9) Scale evolution and entropy decomposition
As the scale s → ∞ s \to \infty s → ∞ E n t μ s ( f s ) \mathop{\mathrm{Ent}}_{\boldsymbol\mu_s}(f_s) Ent μ s ( f s ) f μ f\boldsymbol\mu f μ μ \boldsymbol\mu μ E n t μ ( f ) \mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f) Ent μ ( f ) 4.1 , by the scale evolution of f s f_s f s μ s \boldsymbol\mu_s μ s 4.2 ). In the proofs below, we mainly consider regular densities relative to μ \boldsymbol\mu μ
f ∈ C b ∞ ( R 2 d ) , E μ f = 1 , inf f > 0. \begin{equation}
f\in C_b^\infty(\mathbb R^{2d}),\qquad \mathbb E_{\boldsymbol\mu} f=1,\qquad \inf f>0. \tag*{(4.10)}
\end{equation} f ∈ C b ∞ ( R 2 d ) , E μ f = 1 , inf f > 0. (4.10) The extension to general densities is discussed in Appendix § 7 .
f f f f s = P s f f_s=\mathsf P_sf f s = P s f ∇ y log f s \nabla_y\log f_s ∇ y log f s ∇ w log f s \nabla_w\log f_s ∇ w log f s A A A f s f_s f s μ s \boldsymbol\mu_s μ s
I s y = ∫ f s ∣ ∇ y log f s ∣ 2 d μ s , I s w = ∫ f s ∣ ∇ w log f s ∣ 2 d μ s , I s w , A = ∫ f s ( ∇ w log f s ) ⊤ A ∇ w log f s d μ s , \begin{equation}
\begin{gathered}
\mathcal I^y_s=\int f_s\left\lvert \nabla_y\log f_s\right\rvert^2\,\mathrm d\boldsymbol\mu_s, \qquad \mathcal I^w_s=\int f_s\left\lvert \nabla_w\log f_s\right\rvert^2\,\mathrm d\boldsymbol\mu_s,\\
\mathcal I^{w,A}_s=\int f_s(\nabla_w\log f_s)^\top A\,\nabla_w\log f_s\,\mathrm d\boldsymbol\mu_s,
\end{gathered} \tag*{(4.11)}
\end{equation} I s y = ∫ f s ∣ ∇ y log f s ∣ 2 d μ s , I s w = ∫ f s ∣ ∇ w log f s ∣ 2 d μ s , I s w , A = ∫ f s ( ∇ w log f s ) ⊤ A ∇ w log f s d μ s , (4.11) respectively. We write I s y ( f s ) \mathcal I^y_s(f_s) I s y ( f s ) I s w ( f s ) \mathcal I^w_s(f_s) I s w ( f s ) I s w , A ( f s ) \mathcal I^{w,A}_s(f_s) I s w , A ( f s ) f 0 = f f_0=f f 0 = f μ 0 = μ \boldsymbol\mu_0=\boldsymbol\mu μ 0 = μ I 0 w = I p ( f ) \mathcal I^w_0=\mathcal I_p(f) I 0 w = I p ( f ) (2.6) ). The assumption A ⪰ I d A\succeq\mathrm{Id} A ⪰ Id I s w , A ≥ I s w \mathcal I^{w,A}_s\geq\mathcal I^w_s I s w , A ≥ I s w
Lemma 4.1 (Polchinski generator and renormalized phase-space measure). Define L s = 1 2 Δ y + 1 2 A : ∇ w 2 − ( 1 2 A y + Q s ∇ V s ( Q s y ) ) ⋅ ∇ y − 1 2 A w ⋅ ∇ w .
\mathsf L_s =\frac12\Delta_y +\frac12A:\nabla_w^2 -\left(\frac12Ay+Q_s\nabla V_s(Q_sy)\right)\cdot\nabla_y -\frac12Aw\cdot\nabla_w.
L s = 2 1 Δ y + 2 1 A : ∇ w 2 − ( 2 1 A y + Q s ∇ V s ( Q s y ) ) ⋅ ∇ y − 2 1 A w ⋅ ∇ w . ψ ∈ C b ∞ ( R 2 d ) \psi\in C_b^\infty(\mathbb R^{2d}) ψ ∈ C b ∞ ( R 2 d ) ∂ s ( P s ψ ) = L s ( P s ψ ) \partial_s(\mathsf P_s\psi)=\mathsf L_s(\mathsf P_s\psi) ∂ s ( P s ψ ) = L s ( P s ψ ) f f f f s = P s f f_s = \mathsf P_s f f s = P s f ∂ s f s = L s f s .
\partial_sf_s=\mathsf L_sf_s.
∂ s f s = L s f s .
\frac\,\mathrm d{\,\mathrm ds}\boldsymbol\mu_s
=-\partial_s[V_s(Q_sy)]\,\boldsymbol\mu_s
=-\frac12\operatorname{div}_y\bigl(Q_s\nabla V_s(Q_sy)\,\boldsymbol\mu_s\bigr).
If ψ ∈ C 1 , 2 ( [ 0 , ∞ ) × R 2 d ) \psi\in C^{1,2}([0,\infty)\times\mathbb R^{2d}) ψ ∈ C 1 , 2 ([ 0 , ∞ ) × R 2 d ) ∂ s ψ s \partial_s\psi_s ∂ s ψ s ∇ ( y , w ) k ψ s \nabla_{(y,w)}^k\psi_s ∇ ( y , w ) k ψ s ( 0 ≤ k ≤ 2 ) (0\leq k\leq2) ( 0 ≤ k ≤ 2 ) s s s
\frac\,\mathrm d{\,\mathrm ds}\int \psi_s\,\mathrm d\boldsymbol\mu_s =\int(\partial_s\psi_s-\mathsf L_s\psi_s)\,\mathrm d\boldsymbol\mu_s.
Lemma 4.1 is [BB Proposition 2.1] (see also [BBD2024 Proposition 3.5]) applied to the covariance decomposition (4.2) of C ∞ \mathbf C_\infty C ∞ ( q , p ) ↦ V 0 ( q ) (q,p)\mapsto V_0(q) ( q , p ) ↦ V 0 ( q ) § 8 carries out the change of variables through Q s \mathbf Q_s Q s BB ].
Proof. The coarse-graining operator (4.7) and the density Z 0 − 1 e − V s ( Q s y ) Z_0^{-1}e^{-V_s(Q_sy)} Z 0 − 1 e − V s ( Q s y ) μ s \boldsymbol\mu_s μ s N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ ) N ( Q s ( y , w ) ⊤ , C s ) \mathsf N(\mathbf Q_s(y,w)^\top,\mathbf C_s) N ( Q s ( y , w ) ⊤ , C s ) s ≥ 0 s\geq0 s ≥ 0 Q s + u = Q s Q u \mathbf Q_{s+u}=\mathbf Q_s\mathbf Q_u Q s + u = Q s Q u C s + u = C s + Q s C u Q s \mathbf C_{s+u}=\mathbf C_s+\mathbf Q_s\mathbf C_u\mathbf Q_s C s + u = C s + Q s C u Q s \frac\,\mathrm d{\,\mathrm ds}\mathbf Q_s\big|_{s=0}(y,w)^\top
=-\frac12\operatorname{diag}(A,A)(y,w)^\top , since Q s = e − s A / 2 Q_s=e^{-sA/2} Q s = e − s A /2 \frac\,\mathrm d{\,\mathrm ds}\mathbf C_s\big|_{s=0}=\operatorname{diag}(\mathrm{Id},A) by (4.2) , so it is the Ornstein–Uhlenbeck operator
G = 1 2 Δ y + 1 2 A : ∇ w 2 − 1 2 A y ⋅ ∇ y − 1 2 A w ⋅ ∇ w . \mathcal G=\frac12\Delta_y+\frac12A:\nabla_w^2
-\frac12Ay\cdot\nabla_y-\frac12Aw\cdot\nabla_w. G = 2 1 Δ y + 2 1 A : ∇ w 2 − 2 1 A y ⋅ ∇ y − 2 1 A w ⋅ ∇ w . The Gaussian measure N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ ) G \mathcal G G s s s μ s \boldsymbol\mu_s μ s G \mathcal G G
We first relate L s \mathsf L_s L s G \mathcal G G e − V s ( Q s y ) e^{-V_s(Q_sy)} e − V s ( Q s y ) y y y ∇ y e − V s ( Q s y ) = − e − V s ( Q s y ) Q s ∇ V s ( Q s y ) \nabla_y e^{-V_s(Q_sy)}=-e^{-V_s(Q_sy)}Q_s\nabla V_s(Q_sy) ∇ y e − V s ( Q s y ) = − e − V s ( Q s y ) Q s ∇ V s ( Q s y ) G \mathcal G G ψ ∈ C 2 ( R 2 d ) \psi\in C^2(\mathbb R^{2d}) ψ ∈ C 2 ( R 2 d )
L s ψ = e V s ( Q s y ) G ( e − V s ( Q s y ) ψ ) − e V s ( Q s y ) ψ G ( e − V s ( Q s y ) ) , \begin{equation}
\mathsf L_s\psi
= e^{V_s(Q_sy)} \mathcal G\bigl(e^{-V_s(Q_sy)}\psi\bigr)
- e^{V_s(Q_sy)} \psi\,\mathcal G\bigl(e^{-V_s(Q_sy)}\bigr), \tag*{(4.16)}
\end{equation} L s ψ = e V s ( Q s y ) G ( e − V s ( Q s y ) ψ ) − e V s ( Q s y ) ψ G ( e − V s ( Q s y ) ) , (4.16) that is, L s \mathsf L_s L s G \mathcal G G e − V s ( Q s y ) e^{-V_s(Q_sy)} e − V s ( Q s y )
The equation for P s ψ \mathsf P_s\psi P s ψ ψ ∈ C b ∞ ( R 2 d ) \psi\in C_b^\infty(\mathbb R^{2d}) ψ ∈ C b ∞ ( R 2 d ) (4.7) and (2.12) evaluated at Q s y Q_sy Q s y P s ψ = n s / e − V s ( Q s y ) \mathsf P_s\psi=n_s/e^{-V_s(Q_sy)} P s ψ = n s / e − V s ( Q s y )
n s ( y , w ) = E [ e − V 0 ( Q s y + ζ q ) ψ ( Q s y + ζ q , Q s w + ζ p ) ] , e − V s ( Q s y ) = E [ e − V 0 ( Q s y + ζ q ) ] , n_s(y,w)=\mathbb E\bigl[e^{-V_0(Q_sy+\zeta_q)}
\psi(Q_sy+\zeta_q,Q_sw+\zeta_p)\bigr],
\qquad
e^{-V_s(Q_sy)}=\mathbb E\bigl[e^{-V_0(Q_sy+\zeta_q)}\bigr], n s ( y , w ) = E [ e − V 0 ( Q s y + ζ q ) ψ ( Q s y + ζ q , Q s w + ζ p ) ] , e − V s ( Q s y ) = E [ e − V 0 ( Q s y + ζ q ) ] , with ( ζ q , ζ p ) ∼ N ( 0 , C s ) (\zeta_q,\zeta_p)\sim\mathsf N(0,\mathbf C_s) ( ζ q , ζ p ) ∼ N ( 0 , C s ) e − V 0 ( q ) ψ ( q , p ) e^{-V_0(q)}\psi(q,p) e − V 0 ( q ) ψ ( q , p ) e − V 0 ( q ) e^{-V_0(q)} e − V 0 ( q ) C b ∞ ( R 2 d ) C_b^\infty(\mathbb R^{2d}) C b ∞ ( R 2 d )
∂ s n s = G n s , ∂ s e − V s ( Q s y ) = G ( e − V s ( Q s y ) ) . \begin{equation}
\partial_sn_s=\mathcal G n_s,\qquad
\partial_s e^{-V_s(Q_sy)}=\mathcal G\bigl(e^{-V_s(Q_sy)}\bigr). \tag*{(4.17)}
\end{equation} ∂ s n s = G n s , ∂ s e − V s ( Q s y ) = G ( e − V s ( Q s y ) ) . (4.17) Since n s = e − V s ( Q s y ) P s ψ n_s=e^{-V_s(Q_sy)}\mathsf P_s\psi n s = e − V s ( Q s y ) P s ψ (4.16) give
∂ s ( P s ψ ) = e V s ( Q s y ) ( ∂ s n s − P s ψ ∂ s e − V s ( Q s y ) ) = e V s ( Q s y ) ( G ( e − V s ( Q s y ) P s ψ ) − P s ψ G ( e − V s ( Q s y ) ) ) = L s ( P s ψ ) . \begin{align*}
\partial_s(\mathsf P_s\psi)
&=e^{V_s(Q_sy)}\bigl(\partial_sn_s
-\mathsf P_s\psi\,\partial_se^{-V_s(Q_sy)}\bigr)\\
&=e^{V_s(Q_sy)}\Bigl(\mathcal G\bigl(e^{-V_s(Q_sy)}\mathsf P_s\psi\bigr)
-\mathsf P_s\psi\,\mathcal G\bigl(e^{-V_s(Q_sy)}\bigr)\Bigr)
=\mathsf L_s(\mathsf P_s\psi).
\end{align*} ∂ s ( P s ψ ) = e V s ( Q s y ) ( ∂ s n s − P s ψ ∂ s e − V s ( Q s y ) ) = e V s ( Q s y ) ( G ( e − V s ( Q s y ) P s ψ ) − P s ψ G ( e − V s ( Q s y ) ) ) = L s ( P s ψ ) . This proves (4.13) with ψ = f \psi=f ψ = f
We now derive the measure equation (4.14) , by testing against ψ ∈ C c ∞ ( R 2 d ) \psi\in C_c^\infty(\mathbb R^{2d}) ψ ∈ C c ∞ ( R 2 d ) (4.14) is the derivative of the density Z 0 − 1 e − V s ( Q s y ) Z_0^{-1}e^{-V_s(Q_sy)} Z 0 − 1 e − V s ( Q s y ) μ s \boldsymbol\mu_s μ s N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ ) (4.6) ). For the second, we integrate G \mathcal G G N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ ) e − 1 2 ⟨ y , A y ⟩ − 1 2 ∣ w ∣ 2 e^{-\frac12\left\langle y,Ay\right\rangle-\frac12|w|^2} e − 2 1 ⟨ y , A y ⟩ − 2 1 ∣ w ∣ 2 ψ \psi ψ χ ∈ C 2 ( R 2 d ) \chi\in C^2(\mathbb R^{2d}) χ ∈ C 2 ( R 2 d )
∫ ψ G χ d N ( 0 , C ∞ ) = − 1 2 ∫ ( ∇ y ψ ⋅ ∇ y χ + ( ∇ w ψ ) ⊤ A ∇ w χ ) d N ( 0 , C ∞ ) . \begin{equation}
\int\psi\,\mathcal G\chi\,\mathrm d\mathsf N(0,\mathbf C_\infty)
=-\frac12\int\bigl(\nabla_y\psi\cdot\nabla_y\chi
+(\nabla_w\psi)^\top A\nabla_w\chi\bigr)\,\mathrm d\mathsf N(0,\mathbf C_\infty). \tag*{(4.18)}
\end{equation} ∫ ψ G χ d N ( 0 , C ∞ ) = − 2 1 ∫ ( ∇ y ψ ⋅ ∇ y χ + ( ∇ w ψ ) ⊤ A ∇ w χ ) d N ( 0 , C ∞ ) . (4.18) For χ = e − V s ( Q s y ) \chi=e^{-V_s(Q_sy)} χ = e − V s ( Q s y ) (4.17) gives
\begin{align*}
\frac\,\mathrm d{\,\mathrm ds}\int\psi\,\mathrm d\boldsymbol\mu_s
&=Z_0^{-1}\int\psi\,\mathcal G\bigl(e^{-V_s(Q_sy)}\bigr)
\,\mathrm d\mathsf N(0,\mathbf C_\infty)
=-\frac1{2Z_0}\int\nabla_y\psi\cdot\nabla_ye^{-V_s(Q_sy)}
\,\mathrm d\mathsf N(0,\mathbf C_\infty)\\
&=\frac12\int Q_s\nabla V_s(Q_sy)\cdot\nabla_y\psi
\,\mathrm d\boldsymbol\mu_s,
\end{align*}
which is the second equality tested against ψ \psi ψ
Finally, we integrate (4.16) against N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ ) N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ ) G \mathcal G G (4.17) gives
∫ L s ψ s d μ s = − Z 0 − 1 ∫ ψ s ∂ s e − V s ( Q s y ) d N ( 0 , C ∞ ) . \int\mathsf L_s\psi_s\,\mathrm d\boldsymbol\mu_s
=-Z_0^{-1}\int\psi_s\,\partial_s e^{-V_s(Q_sy)}
\,\mathrm d\mathsf N(0,\mathbf C_\infty). ∫ L s ψ s d μ s = − Z 0 − 1 ∫ ψ s ∂ s e − V s ( Q s y ) d N ( 0 , C ∞ ) . Differentiating under the integral and inserting this identity,
\frac\,\mathrm d{\,\mathrm ds}\int\psi_s\,\mathrm d\boldsymbol\mu_s
=Z_0^{-1}\int\bigl(e^{-V_s(Q_sy)}\partial_s\psi_s
+\psi_s\,\partial_se^{-V_s(Q_sy)}\bigr)\,\mathrm d\mathsf N(0,\mathbf C_\infty)
=\int(\partial_s\psi_s-\mathsf L_s\psi_s)\,\mathrm d\boldsymbol\mu_s,
which is (4.15) . ◻
L s \mathsf L_s L s BGL ], is the bilinear form
Γ s ( ψ , χ ) = 1 2 [ L s ( ψ χ ) − ψ L s χ − χ L s ψ ] , Γ s ( ψ ) = Γ s ( ψ , ψ ) . \begin{equation*}
\Gamma_s(\psi,\chi)=\frac12\bigl[\mathsf L_s(\psi\chi)-\psi\,\mathsf L_s\chi-\chi\,\mathsf L_s\psi\bigr],\qquad \Gamma_s(\psi)=\Gamma_s(\psi,\psi).
\end{equation*} Γ s ( ψ , χ ) = 2 1 [ L s ( ψ χ ) − ψ L s χ − χ L s ψ ] , Γ s ( ψ ) = Γ s ( ψ , ψ ) . Note from (4.12) that only the second-order part 1 2 Δ y + 1 2 A : ∇ w 2 \frac12\Delta_y+\frac12A:\nabla_w^2 2 1 Δ y + 2 1 A : ∇ w 2 L s \mathsf L_s L s Γ s \Gamma_s Γ s
Γ s ( ψ , χ ) = 1 2 ∇ y ψ ⋅ ∇ y χ + 1 2 ( ∇ w ψ ) ⊤ A ∇ w χ , \begin{equation}
\Gamma_s(\psi,\chi)=\frac12\nabla_y\psi\cdot\nabla_y\chi+\frac12(\nabla_w\psi)^\top A\nabla_w\chi, \tag*{(4.19)}
\end{equation} Γ s ( ψ , χ ) = 2 1 ∇ y ψ ⋅ ∇ y χ + 2 1 ( ∇ w ψ ) ⊤ A ∇ w χ , (4.19) and the chain rule
L s [ a ( ψ ) ] = a ′ ( ψ ) L s ψ + a ′ ′ ( ψ ) Γ s ( ψ ) \begin{equation}
\mathsf L_s[a(\psi)]=a'(\psi)\,\mathsf L_s\psi+a''(\psi)\,\Gamma_s(\psi) \tag*{(4.20)}
\end{equation} L s [ a ( ψ )] = a ′ ( ψ ) L s ψ + a ′′ ( ψ ) Γ s ( ψ ) (4.20) holds for every smooth function a : R → R a\colon\mathbb R\to\mathbb R a : R → R Γ s ( f s ) / f s = f s Γ s ( log f s ) \Gamma_s(f_s)/f_s=f_s\Gamma_s(\log f_s) Γ s ( f s ) / f s = f s Γ s ( log f s ) (4.19) and the definitions (4.11) give
∫ Γ s ( f s ) f s d μ s = ∫ f s Γ s ( log f s ) d μ s = 1 2 ( I s y + I s w , A ) . \begin{equation}
\int\frac{\Gamma_s(f_s)}{f_s}\,\mathrm d\boldsymbol\mu_s=\int f_s\Gamma_s(\log f_s)\,\mathrm d\boldsymbol\mu_s=\frac12\bigl(\mathcal I^y_s+\mathcal I^{w,A}_s\bigr). \tag*{(4.21)}
\end{equation} ∫ f s Γ s ( f s ) d μ s = ∫ f s Γ s ( log f s ) d μ s = 2 1 ( I s y + I s w , A ) . (4.21) This is the Fisher information of f s f_s f s μ s \boldsymbol\mu_s μ s Γ s \Gamma_s Γ s
Lemma 4.2 (Entropy decomposition). Let f f f
\frac\,\mathrm d{\,\mathrm ds}\mathop{\mathrm{Ent}}_{\boldsymbol\mu_s}(f_s) =-\int\frac{\Gamma_s(f_s)}{f_s}\,\mathrm d\boldsymbol\mu_s =-\frac12\bigl(\mathcal I^y_s+\mathcal I^{w,A}_s\bigr),
and hence E n t μ ( f ) = 1 2 ∫ 0 ∞ ( I s y + I s w , A ) d s .
\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f)=\frac12\int_0^\infty\bigl(\mathcal I^y_s+\mathcal I^{w,A}_s\bigr)\,\mathrm ds.
Ent μ ( f ) = 2 1 ∫ 0 ∞ ( I s y + I s w , A ) d s .
Proof. By Lemma A.1 , ψ s = f s log f s \psi_s=f_s\log f_s ψ s = f s log f s 4.1 . By (4.15) , (4.13) , and the chain rule (4.20) with a ( u ) = u log u a(u)=u\log u a ( u ) = u log u
\begin{align*}
\frac\,\mathrm d{\,\mathrm ds}\mathop{\mathrm{Ent}}_{\boldsymbol\mu_s}(f_s) &=\int(\partial_s\psi_s-\mathsf L_s\psi_s)\,\mathrm d\boldsymbol\mu_s\\
&=\int\bigl[(1+\log f_s)\mathsf L_sf_s-\mathsf L_s(f_s\log f_s)\bigr]\,\mathrm d\boldsymbol\mu_s =-\int\frac{\Gamma_s(f_s)}{f_s}\,\mathrm d\boldsymbol\mu_s.
\end{align*}
The identity (4.21) gives the second equality in (4.22) . Integrating (4.22) over [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) E n t μ 0 ( f 0 ) = E n t μ ( f ) \mathop{\mathrm{Ent}}_{\boldsymbol\mu_0}(f_0)=\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f) Ent μ 0 ( f 0 ) = Ent μ ( f ) E n t μ s ( f s ) → 0 \mathop{\mathrm{Ent}}_{\boldsymbol\mu_s}(f_s)\to0 Ent μ s ( f s ) → 0 s → ∞ s\to\infty s → ∞ A.4 ), proves (4.23) . ◻
The force defect and the coarse kinetic equation
In this subsection, we identify the evolution of the observation law in kinetic time (the lower row of Figure 1 ). For the free dynamics, the matched observation of § 4.1 makes the diagram commute. With an interaction, the observation law evolves by the density equation of the underdamped Langevin dynamics (2.3) in the coarse variables, with V 0 V_0 V 0 V s ( Q s y ) V_s(Q_sy) V s ( Q s y ) (?) .
( q , p ) (q,p) ( q , p ) f μ f\boldsymbol\mu f μ f f f § 4.1 , the joint law of ( q , p ) (q,p) ( q , p ) f ( q , p ) π s f(q,p)\,\boldsymbol\pi_s f ( q , p ) π s π s \boldsymbol\pi_s π s (4.5) and ( y , w ) (y,w) ( y , w ) f s μ s f_s\boldsymbol\mu_s f s μ s ( q , p ) (q,p) ( q , p ) ( y s , w s ) = ( y , w ) (y_s,w_s)=(y,w) ( y s , w s ) = ( y , w ) f / f s f/f_s f / f s
f ( q , p ) f s ( y , w ) π s ( d q d p ∣ y , w ) , \begin{equation}
\frac{f(q,p)}{f_s(y,w)}\,\boldsymbol\pi_s(\,\mathrm dq\,\,\mathrm dp\mid y,w), \tag*{(4.24)}
\end{equation} f s ( y , w ) f ( q , p ) π s ( d q d p ∣ y , w ) , (4.24) and the conditional interaction force under f μ f\boldsymbol\mu f μ
E [ ∇ V 0 ( q ) ∣ ( y s , w s ) = ( y , w ) ] = ∫ ∇ V 0 ( q ) f ( q , p ) f s ( y , w ) π s ( d q d p ∣ y , w ) = P s ( f ∇ V 0 ) ( y , w ) f s ( y , w ) , \begin{equation*}
\mathbb E\bigl[\nabla V_0(q)\mid(y_s,w_s)=(y,w)\bigr]
=\int\nabla V_0(q)\,\frac{f(q,p)}{f_s(y,w)}\,\boldsymbol\pi_s(\,\mathrm dq\,\,\mathrm dp\mid y,w)
=\frac{\mathsf P_s(f\nabla V_0)(y,w)}{f_s(y,w)},
\end{equation*} E [ ∇ V 0 ( q ) ∣ ( y s , w s ) = ( y , w ) ] = ∫ ∇ V 0 ( q ) f s ( y , w ) f ( q , p ) π s ( d q d p ∣ y , w ) = f s ( y , w ) P s ( f ∇ V 0 ) ( y , w ) , where ( q , p ) ∼ f μ (q,p)\sim f\boldsymbol\mu ( q , p ) ∼ f μ P s \mathsf P_s P s π s \boldsymbol\pi_s π s f = 1 f=1 f = 1 P s ( ∇ V 0 ) \mathsf P_s(\nabla V_0) P s ( ∇ V 0 ) (2.12) in q q q (2.14) gives P s ( ∇ V 0 ) = ∇ V s ( Q s y ) \mathsf P_s(\nabla V_0)=\nabla V_s(Q_sy) P s ( ∇ V 0 ) = ∇ V s ( Q s y ) ∇ V s ( Q s y ) \nabla V_s(Q_sy) ∇ V s ( Q s y ) f s f_s f s y y y Q s Q_s Q s ∇ y [ V s ( Q s y ) ] = Q s ∇ V s ( Q s y ) \nabla_y[V_s(Q_sy)]=Q_s\nabla V_s(Q_sy) ∇ y [ V s ( Q s y )] = Q s ∇ V s ( Q s y )
R s = Q s [ P s ( f ∇ V 0 ) − f s ∇ V s ( Q s y ) ] = Q s [ P s ( f ∇ V 0 ) − P s f P s ( ∇ V 0 ) ] . \begin{equation}
R_s=Q_s\bigl[\mathsf P_s(f\nabla V_0)-f_s\,\nabla V_s(Q_sy)\bigr]
=Q_s\bigl[\mathsf P_s(f\nabla V_0)-\mathsf P_sf\,\mathsf P_s(\nabla V_0)\bigr]. \tag*{(4.25)}
\end{equation} R s = Q s [ P s ( f ∇ V 0 ) − f s ∇ V s ( Q s y ) ] = Q s [ P s ( f ∇ V 0 ) − P s f P s ( ∇ V 0 ) ] . (4.25) By the second form of (4.25) , R s R_s R s Q s Q_s Q s f f f ∇ V 0 \nabla V_0 ∇ V 0 π s ( d q d p ∣ y , w ) \boldsymbol\pi_s(\,\mathrm dq\,\,\mathrm dp\mid y,w) π s ( d q d p ∣ y , w ) R s = 0 R_s=0 R s = 0 f = 1 f=1 f = 1 V 0 = 0 V_0=0 V 0 = 0
∇ q ∗ = − div q + ∇ U ⋅ \nabla_q^*=-\operatorname{div}_q+\nabla U\cdot ∇ q ∗ = − div q + ∇ U ⋅ ∇ p ∗ = − div p + p ⋅ \nabla_p^*=-\operatorname{div}_p+p\cdot ∇ p ∗ = − div p + p ⋅ ∇ q \nabla_q ∇ q ∇ p \nabla_p ∇ p L 2 ( μ ) L^2(\boldsymbol\mu) L 2 ( μ )
∇ y ∗ = − div y + ( A y + Q s ∇ V s ( Q s y ) ) ⋅ , ∇ w ∗ = − div w + w ⋅ \begin{equation*}
\nabla_y^*=-\operatorname{div}_y+\bigl(Ay+Q_s\nabla V_s(Q_sy)\bigr)\cdot,\qquad
\nabla_w^*=-\operatorname{div}_w+w\cdot
\end{equation*} ∇ y ∗ = − div y + ( A y + Q s ∇ V s ( Q s y ) ) ⋅ , ∇ w ∗ = − div w + w ⋅ be the adjoints of ∇ y \nabla_y ∇ y ∇ w \nabla_w ∇ w L 2 ( μ s ) L^2(\boldsymbol\mu_s) L 2 ( μ s ) (2.2) and (4.6) . The adjoint of P s : L 2 ( μ ) → L 2 ( μ s ) \mathsf P_s\colon L^2(\boldsymbol\mu)\to L^2(\boldsymbol\mu_s) P s : L 2 ( μ ) → L 2 ( μ s ) P s ∗ ψ ( q , p ) = E ξ s ψ ( Q s ( q , p ) ⊤ + ξ s ) \mathsf P_s^*\psi(q,p)=\mathbb E_{\boldsymbol\xi_s}\psi\bigl(\mathbf Q_s(q,p)^\top+\boldsymbol\xi_s\bigr) P s ∗ ψ ( q , p ) = E ξ s ψ ( Q s ( q , p ) ⊤ + ξ s ) ∇ q P s ∗ ψ = Q s P s ∗ ( ∇ y ψ ) \nabla_q\mathsf P_s^*\psi=Q_s\mathsf P_s^*(\nabla_y\psi) ∇ q P s ∗ ψ = Q s P s ∗ ( ∇ y ψ ) ∇ p P s ∗ ψ = Q s P s ∗ ( ∇ w ψ ) \nabla_p\mathsf P_s^*\psi=Q_s\mathsf P_s^*(\nabla_w\psi) ∇ p P s ∗ ψ = Q s P s ∗ ( ∇ w ψ )
P s ∇ q ∗ = ∇ y ∗ Q s P s , P s ∇ p ∗ = ∇ w ∗ Q s P s . \begin{equation}
\mathsf P_s\nabla_q^*=\nabla_y^*Q_s\mathsf P_s,\qquad
\mathsf P_s\nabla_p^*=\nabla_w^*Q_s\mathsf P_s. \tag*{(4.26)}
\end{equation} P s ∇ q ∗ = ∇ y ∗ Q s P s , P s ∇ p ∗ = ∇ w ∗ Q s P s . (4.26) The corresponding relations for the gradients follow by differentiating (4.7) with (4.25) :
∇ w P s f = Q s P s ( ∇ p f ) , ∇ y P s f = Q s P s ( ∇ q f ) − R s . \begin{equation}
\nabla_w \mathsf P_s f =Q_s\mathsf P_s(\nabla_pf),\qquad
\nabla_y \mathsf P_s f = Q_s\mathsf P_s(\nabla_qf)-R_s. \tag*{(4.27)}
\end{equation} ∇ w P s f = Q s P s ( ∇ p f ) , ∇ y P s f = Q s P s ( ∇ q f ) − R s . (4.27) We hence see that the interaction force enters the kinetic generator through ∇ q ∗ \nabla_q^* ∇ q ∗ P s ∗ \mathsf P_s^* P s ∗ ∇ q \nabla_q ∇ q e − V 0 e^{-V_0} e − V 0 (4.7) , and the resulting mismatch between ∇ V 0 \nabla V_0 ∇ V 0 ∇ V s ( Q s y ) \nabla V_s(Q_sy) ∇ V s ( Q s y )
(4.6) that μ s \boldsymbol\mu_s μ s 1 2 ⟨ y , A y ⟩ + V s ( Q s y ) + 1 2 ∣ w ∣ 2 \frac12\left\langle y,Ay\right\rangle+V_s(Q_sy)+\frac12|w|^2 2 1 ⟨ y , A y ⟩ + V s ( Q s y ) + 2 1 ∣ w ∣ 2
A s = w ⋅ ∇ y − ( A y + Q s ∇ V s ( Q s y ) ) ⋅ ∇ w , S w = Δ w − w ⋅ ∇ w , \begin{equation}
\mathcal A_s =w\cdot\nabla_y-\bigl(Ay+Q_s\nabla V_s(Q_sy)\bigr)\cdot\nabla_w, \qquad \mathcal S_w=\Delta_w-w\cdot\nabla_w, \tag*{(4.28)}
\end{equation} A s = w ⋅ ∇ y − ( A y + Q s ∇ V s ( Q s y ) ) ⋅ ∇ w , S w = Δ w − w ⋅ ∇ w , (4.28) as in (2.4) . In divergence form, we can write
In particular, A s ∗ = − A s \mathcal A_s^*=-\mathcal A_s A s ∗ = − A s S w ∗ = S w \mathcal S_w^*=\mathcal S_w S w ∗ = S w L 2 ( μ s ) L^2(\boldsymbol\mu_s) L 2 ( μ s )
For g t = P t ∗ g 0 g_t=\mathcal P_t^*g_0 g t = P t ∗ g 0 g t , s = P s g t g_{t,s}=\mathsf P_sg_t g t , s = P s g t (4.9) , the force defect (4.25) of f = g t f=g_t f = g t
R t , s = Q s [ P s ( g t ∇ V 0 ) − g t , s ∇ V s ( Q s y ) ] . \begin{equation*}
R_{t,s}=Q_s\bigl[\mathsf P_s(g_t\nabla V_0)-g_{t,s}\,\nabla V_s(Q_sy)\bigr].
\end{equation*} R t , s = Q s [ P s ( g t ∇ V 0 ) − g t , s ∇ V s ( Q s y ) ] . We also call r t , s = R t , s / g t , s r_{t,s}=R_{t,s}/g_{t,s} r t , s = R t , s / g t , s S w = − ∇ w ∗ ∇ w \mathcal S_w=-\nabla_w^*\nabla_w S w = − ∇ w ∗ ∇ w − ∇ w ∗ R t , s = − ∇ w ∗ ( g t , s r t , s ) -\nabla_w^*R_{t,s}=-\nabla_w^*(g_{t,s}r_{t,s}) − ∇ w ∗ R t , s = − ∇ w ∗ ( g t , s r t , s ) γ S w g t , s = − ∇ w ∗ ( g t , s γ ∇ w log g t , s ) \gamma\mathcal S_wg_{t,s}=-\nabla_w^*(g_{t,s}\,\gamma\nabla_w\log g_{t,s}) γ S w g t , s = − ∇ w ∗ ( g t , s γ ∇ w log g t , s )
Lemma (Matched kinetic intertwining). Let g 0 g_0 g 0 g t = P t ∗ g 0 g_t=\mathcal P_t^*g_0 g t = P t ∗ g 0 s ≥ 0 s\geq0 s ≥ 0 ∂ t g t , s = − A s g t , s + γ S w g t , s − ∇ w ∗ R t , s .
\partial_tg_{t,s} =-\mathcal A_sg_{t,s} +\gamma\mathcal S_wg_{t,s} -\nabla_w^*R_{t,s}.
∂ t g t , s = − A s g t , s + γ S w g t , s − ∇ w ∗ R t , s .
Proof. By Lemma A.2 , g t g_t g t t ≥ 0 t\geq0 t ≥ 0 f f f (4.29) and the identities (4.26) and (4.27) , we have
P s ( A f ) = ∇ w ∗ Q s P s ( ∇ q f ) − ∇ y ∗ Q s P s ( ∇ p f ) = ∇ w ∗ ( ∇ y f s + R s ) − ∇ y ∗ ∇ w f s = A s f s + ∇ w ∗ R s , P s ( S f ) = − ∇ w ∗ Q s P s ( ∇ p f ) = − ∇ w ∗ ∇ w f s = S w f s . \begin{align*}
\mathsf P_s(\mathcal Af)
&=\nabla_w^*Q_s\mathsf P_s(\nabla_qf)-\nabla_y^*Q_s\mathsf P_s(\nabla_pf)
=\nabla_w^*(\nabla_yf_s+R_s)-\nabla_y^*\nabla_wf_s
=\mathcal A_sf_s+\nabla_w^*R_s,\\
\mathsf P_s(\mathcal Sf)
&=-\nabla_w^*Q_s\mathsf P_s(\nabla_pf)
=-\nabla_w^*\nabla_wf_s=\mathcal S_wf_s.
\end{align*} P s ( A f ) P s ( S f ) = ∇ w ∗ Q s P s ( ∇ q f ) − ∇ y ∗ Q s P s ( ∇ p f ) = ∇ w ∗ ( ∇ y f s + R s ) − ∇ y ∗ ∇ w f s = A s f s + ∇ w ∗ R s , = − ∇ w ∗ Q s P s ( ∇ p f ) = − ∇ w ∗ ∇ w f s = S w f s . Since P s \mathsf P_s P s t t t ∂ t g t , s = P s ( ∂ t g t ) \partial_tg_{t,s}=\mathsf P_s(\partial_tg_t) ∂ t g t , s = P s ( ∂ t g t ) f = g t f=g_t f = g t ∂ t g t = − A g t + γ S g t \partial_tg_t=-\mathcal Ag_t+\gamma\mathcal Sg_t ∂ t g t = − A g t + γ S g t (2.5) , we obtain
\begin{equation*}
\partial_tg_{t,s}
=-\mathsf P_s(\mathcal Ag_t)+\gamma\mathsf P_s(\mathcal Sg_t)
=-\mathcal A_sg_{t,s}+\gamma\mathcal S_wg_{t,s}-\nabla_w^*R_{t,s}.\qedhere
\end{equation*}
◻
The corrector and entropy decay
This section is devoted to the proofs of Theorem 3.1 and Corollary 3.3 . We add to the entropy a corrector, built from the entropy decomposition of Lemma 4.2 , and reduce both results to a bound on the kinetic derivative of the corrector (Proposition 5.2 ). This bound rests on the exact identity for the kinetic derivative of the cross term at each scale (Lemma 5.3 ), and two energy estimates along the Polchinski flow, one carrying the control of the momentum Fisher information by friction from scale zero to all scales (Lemma 5.4 ), the other bounding the force defect by the entropy (Proposition 5.5 ).
Throughout, g 0 g_0 g 0 (4.10) holds with f = g 0 f=g_0 f = g 0 g t = P t ∗ g 0 g_t=\mathcal P_t^*g_0 g t = P t ∗ g 0 g t , s = P s g t g_{t,s}=\mathsf P_sg_t g t , s = P s g t A.2 , g t g_t g t t ≥ 0 t\geq0 t ≥ 0 f = g t f=g_t f = g t f s = g t , s f_s=g_{t,s} f s = g t , s s s s g t , s g_{t,s} g t , s (4.30) , relative to the reference measure μ s \boldsymbol\mu_s μ s t t t
The corrector and the proof of Theorem 3.1
Following the modified entropy method of hypocoercivity [Villani ], we add to the entropy a term coupling the position and momentum scores, which captures the transfer of dissipation from momentum to position by Hamiltonian transport.
Specifically, let f f f f s = P s f f_s=\mathsf P_sf f s = P s f s s s
J s ( f s ) = ∫ f s ∇ y log f s ⋅ ∇ w log f s d μ s , \begin{equation*}
\mathcal J_s(f_s)=\int f_s\,\nabla_y\log f_s\cdot\nabla_w\log f_s\,\mathrm d\boldsymbol\mu_s,
\end{equation*} J s ( f s ) = ∫ f s ∇ y log f s ⋅ ∇ w log f s d μ s , and the corrector by integrating the cross term over all scales:
J ( f ) = ∫ 0 ∞ J s ( f s ) d s , \begin{equation}
\mathcal J(f)=\int_0^\infty\mathcal J_s(f_s)\,\mathrm ds, \tag*{(5.1)}
\end{equation} J ( f ) = ∫ 0 ∞ J s ( f s ) d s , (5.1) as the entropy decomposition of Lemma 4.2 integrates Fisher informations over scales. Then, we introduce the modified entropy, for 0 < θ < 1 0<\theta<1 0 < θ < 1
E θ ( f ) = E n t μ ( f ) + θ J ( f ) . \begin{equation}
\mathcal E_\theta(f)=\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f)+\theta\,\mathcal J(f). \tag*{(5.2)}
\end{equation} E θ ( f ) = Ent μ ( f ) + θ J ( f ) . (5.2) We first bound the size of the corrector in Lemma 5.1 , which gives the equivalence between the modified entropy E θ ( f ) \mathcal E_\theta(f) E θ ( f ) E n t μ ( f ) \mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f) Ent μ ( f )
( 1 − θ ) E n t μ ( f ) ≤ E θ ( f ) ≤ ( 1 + θ ) E n t μ ( f ) . \begin{equation}
(1-\theta)\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f)\leq\mathcal E_\theta(f)\leq(1+\theta)\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f). \tag*{(5.3)}
\end{equation} ( 1 − θ ) Ent μ ( f ) ≤ E θ ( f ) ≤ ( 1 + θ ) Ent μ ( f ) . (5.3) Lemma 5.1 (Size of the corrector). For a regular density f f f (5.1) is well-defined, and satisfies ∣ J ( f ) ∣ ≤ E n t μ ( f ) .
\left\lvert \mathcal J(f)\right\rvert\leq \mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f).
∣ J ( f ) ∣ ≤ Ent μ ( f ) .
Proof. By Young’s inequality and A ⪰ I d A\succeq\mathrm{Id} A ⪰ Id J s ( f s ) \mathcal J_s(f_s) J s ( f s )
2 ∣ ∇ y log f s ⋅ ∇ w log f s ∣ ≤ ∣ ∇ y log f s ∣ 2 + ∣ ∇ w log f s ∣ 2 ≤ ∣ ∇ y log f s ∣ 2 + ( ∇ w log f s ) ⊤ A ∇ w log f s . \begin{equation*}
2\bigl|\nabla_y\log f_s\cdot\nabla_w\log f_s\bigr|\leq|\nabla_y\log f_s|^2+|\nabla_w\log f_s|^2\leq|\nabla_y\log f_s|^2+(\nabla_w\log f_s)^\top A\,\nabla_w\log f_s.
\end{equation*} 2 ∇ y log f s ⋅ ∇ w log f s ≤ ∣ ∇ y log f s ∣ 2 + ∣ ∇ w log f s ∣ 2 ≤ ∣ ∇ y log f s ∣ 2 + ( ∇ w log f s ) ⊤ A ∇ w log f s . Multiplying by f s f_s f s μ s \boldsymbol\mu_s μ s s s s (4.23) gives
∫ 0 ∞ ∫ f s ∣ ∇ y log f s ⋅ ∇ w log f s ∣ d μ s d s ≤ 1 2 ∫ 0 ∞ ( I s y + I s w , A ) d s = E n t μ ( f ) . \begin{equation*}
\int_0^\infty\!\int f_s\bigl|\nabla_y\log f_s\cdot\nabla_w\log f_s\bigr|\,\mathrm d\boldsymbol\mu_s\,\mathrm ds\leq\frac12\int_0^\infty\bigl(\mathcal I^y_s+\mathcal I^{w,A}_s\bigr)\,\mathrm ds=\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f).
\end{equation*} ∫ 0 ∞ ∫ f s ∇ y log f s ⋅ ∇ w log f s d μ s d s ≤ 2 1 ∫ 0 ∞ ( I s y + I s w , A ) d s = Ent μ ( f ) . This proves (5.4) , completing the proof. ◻
We now prove Theorem 3.1 by showing that the modified entropy E θ ( g t ) \mathcal E_\theta(g_t) E θ ( g t ) θ \theta θ 3.3 by linearization. By (2.7) , the friction part dissipates entropy only through the momentum Fisher information. The time derivative of J ( g t ) \mathcal J(g_t) J ( g t )
Proposition 5.2 (Bound on the kinetic derivative of the corrector). Let g t = P t ∗ g 0 g_t=\mathcal P_t^*g_0 g t = P t ∗ g 0 g 0 g_0 g 0
\begin{aligned}
\frac\,\mathrm d{\,\mathrm dt}\mathcal J(g_t) \leq{}& -2(1-\eta)\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t) +\left(\max\{2,\gamma\}+\frac{\eta}{\sqrt2}\right)\mathcal I_p(g_t)\\
&+\sqrt2(\gamma+\eta)\sqrt{\mathcal I_p(g_t)\,\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)}.
\end{aligned}
The first term on the right-hand side of (5.5) provides dissipation whenever η < 1 \eta<1 η < 1 η \eta η θ \theta θ γ I p ( g t ) \gamma\mathcal I_p(g_t) γ I p ( g t ) 5.2 to § 5.2 and first deduce Theorem 3.1 and Corollary 3.3 from it.
Proof of Theorem 3.1 . Let g 0 g_0 g 0 g t = P t ∗ g 0 g_t=\mathcal P_t^*g_0 g t = P t ∗ g 0 θ = θ γ , η ∗ \theta=\theta_{\gamma,\eta_*} θ = θ γ , η ∗ (3.4) . Since η ≤ η ∗ < 1 \eta\leq\eta_*<1 η ≤ η ∗ < 1 (2.7) and Proposition 5.2 give
\begin{align*}
-\frac\,\mathrm d{\,\mathrm dt}\mathcal E_\theta(g_t) &\geq (\gamma-\theta K_{\gamma,\eta_*})\mathcal I_p(g_t)+2\theta(1-\eta_*)\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t) -\theta\sqrt2(\gamma+\eta_*)\sqrt{\mathcal I_p(g_t)\,\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)}\\
&\geq\frac\gamma2\mathcal I_p(g_t)+2\theta(1-\eta_*)\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t) -\theta\sqrt2(\gamma+\eta_*)\sqrt{\mathcal I_p(g_t)\,\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)}.
\end{align*}
The last line uses θ ≤ γ / ( 2 K γ , η ∗ ) \theta\leq\gamma/(2K_{\gamma,\eta_*}) θ ≤ γ / ( 2 K γ , η ∗ ) (3.4) . Young’s inequality and θ ≤ γ ( 1 − η ∗ ) / ( 2 ( γ + η ∗ ) 2 ) \theta\leq\gamma(1-\eta_*)/\bigl(2(\gamma+\eta_*)^2\bigr) θ ≤ γ ( 1 − η ∗ ) / ( 2 ( γ + η ∗ ) 2 ) (3.4) yield
θ 2 ( γ + η ∗ ) I p ( g t ) E n t μ ( g t ) ≤ γ 4 I p ( g t ) + 2 θ 2 ( γ + η ∗ ) 2 γ E n t μ ( g t ) ≤ γ 4 I p ( g t ) + θ ( 1 − η ∗ ) E n t μ ( g t ) . \begin{equation*}
\theta\sqrt2(\gamma+\eta_*)\sqrt{\mathcal I_p(g_t)\,\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)} \leq\frac\gamma4\mathcal I_p(g_t) +\frac{2\theta^2(\gamma+\eta_*)^2}{\gamma}\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t) \leq\frac\gamma4\mathcal I_p(g_t)+\theta(1-\eta_*)\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t).
\end{equation*} θ 2 ( γ + η ∗ ) I p ( g t ) Ent μ ( g t ) ≤ 4 γ I p ( g t ) + γ 2 θ 2 ( γ + η ∗ ) 2 Ent μ ( g t ) ≤ 4 γ I p ( g t ) + θ ( 1 − η ∗ ) Ent μ ( g t ) . It follows that
\begin{equation}
-\frac\,\mathrm d{\,\mathrm dt}\mathcal E_\theta(g_t) \geq\frac\gamma4\mathcal I_p(g_t)+\theta(1-\eta_*)\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t) \geq c_{\gamma,\eta_*}\mathcal E_\theta(g_t). \tag*{(5.6)}
\end{equation}
The last inequality uses E θ ( g t ) ≤ ( 1 + θ ) E n t μ ( g t ) \mathcal E_\theta(g_t)\leq(1+\theta)\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t) E θ ( g t ) ≤ ( 1 + θ ) Ent μ ( g t ) (5.3) and the definition of c γ , η ∗ c_{\gamma,\eta_*} c γ , η ∗ (3.3) . Then, by Gronwall’s inequality and (5.3) , we have
E n t μ ( g t ) ≤ E θ ( g t ) 1 − θ ≤ e − c γ , η ∗ t E θ ( g 0 ) 1 − θ ≤ C γ , η ∗ e − c γ , η ∗ t E n t μ ( g 0 ) , \begin{equation*}
\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)\leq\frac{\mathcal E_\theta(g_t)}{1-\theta} \leq\frac{e^{-c_{\gamma,\eta_*}t}\,\mathcal E_\theta(g_0)}{1-\theta} \leq C_{\gamma,\eta_*}e^{-c_{\gamma,\eta_*}t}\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_0),
\end{equation*} Ent μ ( g t ) ≤ 1 − θ E θ ( g t ) ≤ 1 − θ e − c γ , η ∗ t E θ ( g 0 ) ≤ C γ , η ∗ e − c γ , η ∗ t Ent μ ( g 0 ) , which is (3.2) for regular densities. The approximation argument in Appendix § 7 extends the estimate to every finite-entropy initial density. ◻
Proof of Corollary 3.3 . Let ψ \psi ψ ∣ κ ∣ |\kappa| ∣ κ ∣ 3.1 applied to 1 + κ ψ 1+\kappa\psi 1 + κ ψ P t ∗ ( 1 + κ ψ ) = 1 + κ P t ∗ ψ \mathcal P_t^*(1+\kappa\psi)=1+\kappa\mathcal P_t^*\psi P t ∗ ( 1 + κ ψ ) = 1 + κ P t ∗ ψ
E n t μ ( 1 + κ P t ∗ ψ ) ≤ C γ , η ∗ e − c γ , η ∗ t E n t μ ( 1 + κ ψ ) . \begin{equation*}
\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(1+\kappa\mathcal P_t^*\psi)\leq C_{\gamma,\eta_*}e^{-c_{\gamma,\eta_*}t}\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(1+\kappa\psi).
\end{equation*} Ent μ ( 1 + κ P t ∗ ψ ) ≤ C γ , η ∗ e − c γ , η ∗ t Ent μ ( 1 + κ ψ ) . Since ∥ P t ∗ ψ ∥ ∞ ≤ ∥ ψ ∥ ∞ \left\lVert \mathcal P_t^*\psi\right\rVert_\infty\le\left\lVert \psi\right\rVert_\infty ∥ P t ∗ ψ ∥ ∞ ≤ ∥ ψ ∥ ∞ E n t μ ( 1 + κ ψ ) = κ 2 2 ∥ ψ ∥ 2 2 + o ( κ 2 ) \mathop{\mathrm{Ent}}_{\boldsymbol\mu}(1+\kappa\psi)=\frac{\kappa^2}{2}\left\lVert \psi\right\rVert_2^2+o(\kappa^2) Ent μ ( 1 + κ ψ ) = 2 κ 2 ∥ ψ ∥ 2 2 + o ( κ 2 ) κ 2 / 2 \kappa^2/2 κ 2 /2 κ → 0 \kappa\to0 κ → 0
∥ P t ∗ ψ ∥ 2 2 ≤ C γ , η ∗ e − c γ , η ∗ t ∥ ψ ∥ 2 2 . \begin{equation*}
\left\lVert \mathcal P_t^*\psi\right\rVert_2^2\le C_{\gamma,\eta_*} e^{-c_{\gamma,\eta_*}t}\left\lVert \psi\right\rVert_2^2.
\end{equation*} ∥ P t ∗ ψ ∥ 2 2 ≤ C γ , η ∗ e − c γ , η ∗ t ∥ ψ ∥ 2 2 . The standard density argument extends this estimate to centered ψ ∈ L 2 ( μ ) \psi\in L^2(\boldsymbol\mu) ψ ∈ L 2 ( μ ) P t \mathcal P_t P t
Bound on the kinetic derivative of the corrector
To prove Proposition 5.2 , we integrate the kinetic derivative of the cross term over scales and bound the resulting terms by energy estimates along the Polchinski flow, applied with f = g t f=g_t f = g t 5.3 ), proved in § 5.3 , and the momentum and force-defect energy estimates for a fixed density (Lemma 5.4 and Proposition 5.5 ), proved in § 5.4 .
Lemma 5.3 (Exact evolution of the cross term). Let g t = P t ∗ g 0 g_t=\mathcal P_t^*g_0 g t = P t ∗ g 0 g 0 g_0 g 0 s ≥ 0 s\geq0 s ≥ 0 h = log g t , s h=\log g_{t,s} h = log g t , s ∂ t J s ( g t , s ) = − I s y ( g t , s ) + I s w , A ( g t , s ) ⏟ free Hamiltonian contribution + ∫ g t , s ( ∇ w h ) ⊤ Q s ∇ 2 V s ( Q s y ) Q s ∇ w h d μ s ⏟ smoothed interaction Hessian − γ J s ( g t , s ) ⏟ friction, cross term − 2 γ ∫ g t , s ∑ i , j ∂ y i ∂ w j h ∂ w i ∂ w j h d μ s ⏟ friction, mixed Hessians − ∫ g t , s r t , s ⋅ ∇ y h d μ s ⏟ force defect, position score − ∫ g t , s ∑ i , j [ ( ∂ y i r t , s , j ) ∂ w i ∂ w j h + ( ∂ w i r t , s , j ) ∂ y i ∂ w j h ] d μ s ⏟ force defect, derivatives . \begin{aligned}
\partial_t\mathcal J_s(g_{t,s})
={}&\underbrace{-\mathcal I^y_s(g_{t,s})+\mathcal I^{w,A}_s(g_{t,s})}_{\text{free Hamiltonian contribution}}
+\underbrace{\int g_{t,s}(\nabla_wh)^\top Q_s\nabla^2V_s(Q_sy)Q_s\nabla_wh\,\mathrm d\boldsymbol\mu_s}_{\text{smoothed interaction Hessian}}\notag\\
&\underbrace{-\gamma\mathcal J_s(g_{t,s})}_{\text{friction, cross term}}
\;\underbrace{-2\gamma\int g_{t,s}\sum_{i,j}\partial_{y_i}\partial_{w_j}h\,\partial_{w_i}\partial_{w_j}h\,\mathrm d\boldsymbol\mu_s}_{\text{friction, mixed Hessians}}\notag\\
&\underbrace{-\int g_{t,s}\,r_{t,s}\cdot\nabla_yh\,\mathrm d\boldsymbol\mu_s}_{\text{force defect, position score}}
\;\underbrace{-\int g_{t,s}\sum_{i,j}\bigl[(\partial_{y_i}r_{t,s,j})\,\partial_{w_i}\partial_{w_j}h+(\partial_{w_i}r_{t,s,j})\,\partial_{y_i}\partial_{w_j}h\bigr]\,\mathrm d\boldsymbol\mu_s}_{\text{force defect, derivatives}}.
\end{aligned} ∂ t J s ( g t , s ) = free Hamiltonian contribution − I s y ( g t , s ) + I s w , A ( g t , s ) + smoothed interaction Hessian ∫ g t , s ( ∇ w h ) ⊤ Q s ∇ 2 V s ( Q s y ) Q s ∇ w h d μ s friction, cross term − γ J s ( g t , s ) friction, mixed Hessians − 2 γ ∫ g t , s i , j ∑ ∂ y i ∂ w j h ∂ w i ∂ w j h d μ s force defect, position score − ∫ g t , s r t , s ⋅ ∇ y h d μ s force defect, derivatives − ∫ g t , s i , j ∑ [ ( ∂ y i r t , s , j ) ∂ w i ∂ w j h + ( ∂ w i r t , s , j ) ∂ y i ∂ w j h ] d μ s .
The free Hamiltonian contribution in (5.7) comes from the part of the transport − A s -\mathcal A_s − A s 1 2 ⟨ y , A y ⟩ + 1 2 ∣ w ∣ 2 \frac12\left\langle y,Ay\right\rangle+\frac12|w|^2 2 1 ⟨ y , A y ⟩ + 2 1 ∣ w ∣ 2 V s ( Q s y ) V_s(Q_sy) V s ( Q s y ) I s w , A ( g t , s ) \mathcal I^{w,A}_s(g_{t,s}) I s w , A ( g t , s ) A A A Q s ∇ 2 V s ( Q s y ) Q s Q_s\nabla^2V_s(Q_sy)Q_s Q s ∇ 2 V s ( Q s y ) Q s I s w , A ( g t , s ) \mathcal I^{w,A}_s(g_{t,s}) I s w , A ( g t , s ) A + Q s ∇ 2 V s ( Q s y ) Q s A+Q_s\nabla^2V_s(Q_sy)Q_s A + Q s ∇ 2 V s ( Q s y ) Q s 1 2 ⟨ y , A y ⟩ + V s ( Q s y ) \frac12\left\langle y,Ay\right\rangle+V_s(Q_sy) 2 1 ⟨ y , A y ⟩ + V s ( Q s y ) V s = 0 V_s=0 V s = 0 ( A − 1 / 2 ∇ y log g t , s , ∇ w log g t , s ) (A^{-1/2}\nabla_y\log g_{t,s},\nabla_w\log g_{t,s}) ( A − 1/2 ∇ y log g t , s , ∇ w log g t , s ) λ 1 / 2 \lambda^{1/2} λ 1/2 A A A λ \lambda λ I s w , A ( g t , s ) − I s y ( g t , s ) \mathcal I^{w,A}_s(g_{t,s})-\mathcal I^y_s(g_{t,s}) I s w , A ( g t , s ) − I s y ( g t , s ) a b ab ab a ˙ = b \dot a=b a ˙ = b b ˙ = − a \dot b=-a b ˙ = − a (4.23) turns − I s y ( g t , s ) -\mathcal I^y_s(g_{t,s}) − I s y ( g t , s ) − 2 E n t μ ( g t ) -2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t) − 2 Ent μ ( g t ) I s w , A ( g t , s ) \mathcal I^{w,A}_s(g_{t,s}) I s w , A ( g t , s )
The other terms come from friction (the terms carrying γ \gamma γ (4.30) , both enter as momentum divergences − ∇ w ∗ ( g t , s r ) -\nabla_w^*(g_{t,s}r) − ∇ w ∗ ( g t , s r ) r = γ ∇ w log g t , s r=\gamma\nabla_w\log g_{t,s} r = γ ∇ w log g t , s r = r t , s r=r_{t,s} r = r t , s r r r g t , s μ s g_{t,s}\boldsymbol\mu_s g t , s μ s − r -r − r μ s \boldsymbol\mu_s μ s N ( 0 , I d ) \mathsf N(0,\mathrm{Id}) N ( 0 , Id ) ∂ t ∇ w log g t , s = ( r ⋅ ∇ w ) ∇ w log g t , s − r \partial_t\nabla_w\log g_{t,s}=(r\cdot\nabla_w)\nabla_w\log g_{t,s}-r ∂ t ∇ w log g t , s = ( r ⋅ ∇ w ) ∇ w log g t , s − r − γ J s ( g t , s ) -\gamma\mathcal J_s(g_{t,s}) − γ J s ( g t , s ) − ∫ g t , s r t , s ⋅ ∇ y log g t , s d μ s -\int g_{t,s}\,r_{t,s}\cdot\nabla_y\log g_{t,s}\,\mathrm d\boldsymbol\mu_s − ∫ g t , s r t , s ⋅ ∇ y log g t , s d μ s (5.7) .
Recall from (2.7) that friction dissipates entropy only through I p ( g t ) \mathcal I_p(g_t) I p ( g t ) \frac\,\mathrm d{\,\mathrm dt}\mathcal J(g_t) involves the momentum score at every scale. Lemma 5.4 bounds the momentum score with its derivatives, square integrated over all scales, by I p ( f ) \mathcal I_p(f) I p ( f ) 5.5 bounds the normalized force defect with its derivatives in the same way by 2 η 2 E n t μ ( f ) 2\eta^2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f) 2 η 2 Ent μ ( f ) f f f f s = P s f f_s=\mathsf P_sf f s = P s f ψ = ( ψ 1 , … , ψ d ) \psi=(\psi_1,\dots,\psi_d) ψ = ( ψ 1 , … , ψ d )
D s ( ψ ) = 2 ∑ i ∫ f s Γ s ( ψ i ) d μ s = ∫ f s [ ∑ i ∣ ∂ y i ψ ∣ 2 + ∑ i , j A i j ⟨ ∂ w i ψ , ∂ w j ψ ⟩ ] d μ s . \begin{equation}
\mathcal D_s(\psi)=2\sum_i\int f_s\Gamma_s(\psi_i)\,\mathrm d\boldsymbol\mu_s =\int f_s\left[ \sum_i\left\lvert \partial_{y_i}\psi\right\rvert^2 +\sum_{i,j}A_{ij} \left\langle \partial_{w_i}\psi,\partial_{w_j}\psi\right\rangle \right]\,\mathrm d\boldsymbol\mu_s. \tag*{(5.8)}
\end{equation} D s ( ψ ) = 2 i ∑ ∫ f s Γ s ( ψ i ) d μ s = ∫ f s [ i ∑ ∣ ∂ y i ψ ∣ 2 + i , j ∑ A ij ⟨ ∂ w i ψ , ∂ w j ψ ⟩ ] d μ s . (5.8) Lemma 5.4 (Dissipation of momentum Fisher information). For a regular density f f f f s = P s f f_s=\mathsf P_sf f s = P s f
\frac\,\mathrm d{\,\mathrm ds}\mathcal I^w_s=-\mathcal D_s(\nabla_w\log f_s)-\mathcal I^{w,A}_s.
Consequently, I s w ≤ e − s I p ( f ) , s ≥ 0 ,
\mathcal I^w_s\leq e^{-s}\mathcal I_p(f),\qquad s\geq0,
I s w ≤ e − s I p ( f ) , s ≥ 0 , ∫ 0 ∞ ( D s ( ∇ w log f s ) + I s w , A ) d s ≤ I p ( f ) .
\int_0^\infty\bigl(\mathcal D_s(\nabla_w\log f_s)+\mathcal I^{w,A}_s\bigr)\,\mathrm ds\leq \mathcal I_p(f).
∫ 0 ∞ ( D s ( ∇ w log f s ) + I s w , A ) d s ≤ I p ( f ) . s ≥ 0 s\geq0 s ≥ 0 D s ( ∇ w log f s ) ≥ ∫ f s [ ∥ ∇ y ∇ w log f s ∥ H S 2 + ∥ ∇ w 2 log f s ∥ H S 2 ] d μ s ,
\mathcal D_s(\nabla_w\log f_s)\geq\int f_s\bigl[\left\lVert \nabla_y\nabla_w\log f_s\right\rVert_{\mathrm{HS}}^2+\left\lVert \nabla_w^2\log f_s\right\rVert_{\mathrm{HS}}^2\bigr]\,\mathrm d\boldsymbol\mu_s,
D s ( ∇ w log f s ) ≥ ∫ f s [ ∥ ∇ y ∇ w log f s ∥ HS 2 + ∇ w 2 log f s HS 2 ] d μ s , ∫ 0 ∞ ∫ f s [ ∥ ∇ y ∇ w log f s ∥ H S 2 + ∥ ∇ w 2 log f s ∥ H S 2 ] d μ s d s ≤ I p ( f ) .
\int_0^\infty\!\int f_s \left[ \left\lVert \nabla_y\nabla_w\log f_s\right\rVert_{\mathrm{HS}}^2 +\left\lVert \nabla_w^2\log f_s\right\rVert_{\mathrm{HS}}^2 \right]\,\mathrm d\boldsymbol\mu_s\,\mathrm ds \leq \mathcal I_p(f).
∫ 0 ∞ ∫ f s [ ∥ ∇ y ∇ w log f s ∥ HS 2 + ∇ w 2 log f s HS 2 ] d μ s d s ≤ I p ( f ) .
The force defect satisfies an energy estimate of the same form, with the smoothed interaction Hessian as a source.
Proposition 5.5 (Energy estimate for the force defect). Let f f f R s R_s R s (4.25) satisfies ∂ s R s = L s R s − 1 2 A R s + Q s ∇ 2 V s ( Q s y ) Q s ∇ y f s , R 0 = 0.
\partial_sR_s =\mathsf L_sR_s-\frac12AR_s+Q_s\nabla^2V_s(Q_sy)Q_s\nabla_yf_s, \qquad R_0=0.
∂ s R s = L s R s − 2 1 A R s + Q s ∇ 2 V s ( Q s y ) Q s ∇ y f s , R 0 = 0. r s = R s / f s r_s=R_s/f_s r s = R s / f s ∫ 0 ∞ [ D s ( r s ) + ∫ f s r s ⊤ A r s d μ s ] d s ≤ 2 η 2 E n t μ ( f ) .
\int_0^\infty \left[ \mathcal D_s(r_s) +\int f_s r_s^\top Ar_s\,\mathrm d\boldsymbol\mu_s \right]\,\mathrm ds \leq2\eta^2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f).
∫ 0 ∞ [ D s ( r s ) + ∫ f s r s ⊤ A r s d μ s ] d s ≤ 2 η 2 Ent μ ( f ) . ∫ 0 ∞ ∫ f s [ ∥ ∇ y r s ∥ H S 2 + ∥ ∇ w r s ∥ H S 2 + ∣ r s ∣ 2 ] d μ s d s ≤ 2 η 2 E n t μ ( f ) .
\int_0^\infty\!\int f_s \left[ \left\lVert \nabla_yr_s\right\rVert_{\mathrm{HS}}^2 +\left\lVert \nabla_wr_s\right\rVert_{\mathrm{HS}}^2+\left\lvert r_s\right\rvert^2 \right]\,\mathrm d\boldsymbol\mu_s\,\mathrm ds \leq2\eta^2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f).
∫ 0 ∞ ∫ f s [ ∥ ∇ y r s ∥ HS 2 + ∥ ∇ w r s ∥ HS 2 + ∣ r s ∣ 2 ] d μ s d s ≤ 2 η 2 Ent μ ( f ) .
Proof of Proposition 5.2 . Fix t t t f = g t f=g_t f = g t f s = g t , s f_s=g_{t,s} f s = g t , s r s = r t , s r_s=r_{t,s} r s = r t , s I = I p ( g t ) \mathcal I=\mathcal I_p(g_t) I = I p ( g t ) (7.5) , \frac\,\mathrm d{\,\mathrm dt}\mathcal J(g_t)=\int_0^\infty\partial_t\mathcal J_s(g_{t,s})\,\mathrm ds , and we integrate the terms of (5.7) in s s s
For the free Hamiltonian contribution, by (4.23) we have
∫ 0 ∞ ( − I s y + I s w , A ) d s = − 2 E n t μ ( g t ) + 2 ∫ 0 ∞ I s w , A d s . \begin{equation*}
\int_0^\infty(-\mathcal I^y_s+\mathcal I^{w,A}_s)\,\mathrm ds =-2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)+2\int_0^\infty \mathcal I^{w,A}_s\,\mathrm ds.
\end{equation*} ∫ 0 ∞ ( − I s y + I s w , A ) d s = − 2 Ent μ ( g t ) + 2 ∫ 0 ∞ I s w , A d s . For the force-defect term with the position score, the Cauchy–Schwarz inequality, (5.16) , and (4.23) give
∣ ∫ 0 ∞ ∫ f s r s ⋅ ∇ y log f s d μ s d s ∣ ≤ 2 η 2 E n t μ ( g t ) 2 E n t μ ( g t ) = 2 η E n t μ ( g t ) . \begin{equation*}
\left| \int_0^\infty\!\int f_sr_s\cdot\nabla_y\log f_s\,\mathrm d\boldsymbol\mu_s\,\mathrm ds \right| \leq\sqrt{2\eta^2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)}\sqrt{2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)}=2\eta \mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t).
\end{equation*} ∫ 0 ∞ ∫ f s r s ⋅ ∇ y log f s d μ s d s ≤ 2 η 2 Ent μ ( g t ) 2 Ent μ ( g t ) = 2 η Ent μ ( g t ) . For the smoothed interaction Hessian, the definition of b ( s ) b(s) b ( s ) (5.10) , the Cauchy–Schwarz inequality in s s s (3.1) of η \eta η
∣ ∫ 0 ∞ ∫ f s ( ∇ w log f s ) ⊤ Q s ∇ 2 V s ( Q s y ) Q s ∇ w log f s d μ s d s ∣ ≤ ∫ 0 ∞ b ( s ) I s w d s ≤ I ∫ 0 ∞ b ( s ) e − s d s ≤ η 2 I . \begin{align*}
&\left| \int_0^\infty\!\int f_s(\nabla_w\log f_s)^\top Q_s\nabla^2V_s(Q_sy)Q_s\nabla_w\log f_s\,\mathrm d\boldsymbol\mu_s\,\mathrm ds \right|\\
&\qquad\leq\int_0^\infty b(s)\mathcal I^w_s\,\mathrm ds \leq \mathcal I\int_0^\infty b(s)e^{-s}\,\mathrm ds \leq\frac{\eta}{\sqrt2}\mathcal I.
\end{align*} ∫ 0 ∞ ∫ f s ( ∇ w log f s ) ⊤ Q s ∇ 2 V s ( Q s y ) Q s ∇ w log f s d μ s d s ≤ ∫ 0 ∞ b ( s ) I s w d s ≤ I ∫ 0 ∞ b ( s ) e − s d s ≤ 2 η I . For the mixed Hessians, by the Cauchy–Schwarz inequality in the indices i , j i,j i , j (5.12) , we obtain
2 ∣ ∫ f s ∑ i , j ( ∂ y i ∂ w j log f s ) ( ∂ w i ∂ w j log f s ) d μ s ∣ ≤ D s ( ∇ w log f s ) . \begin{equation}
2\left| \int f_s \sum_{i,j}(\partial_{y_i}\partial_{w_j}\log f_s)(\partial_{w_i}\partial_{w_j}\log f_s) \,\mathrm d\boldsymbol\mu_s \right|\leq\mathcal D_s(\nabla_w\log f_s). \tag*{(5.17)}
\end{equation} 2 ∫ f s i , j ∑ ( ∂ y i ∂ w j log f s ) ( ∂ w i ∂ w j log f s ) d μ s ≤ D s ( ∇ w log f s ) . (5.17) We estimate the friction term containing the mixed Hessians together with the momentum term of the free Hamiltonian contribution. By (5.17) and (5.11) ,
2 ∫ 0 ∞ I s w , A d s − 2 γ ∫ 0 ∞ ∫ f s ∑ i , j ( ∂ y i ∂ w j log f s ) ( ∂ w i ∂ w j log f s ) d μ s d s ≤ 2 ∫ 0 ∞ I s w , A d s + γ ∫ 0 ∞ D s ( ∇ w log f s ) d s ≤ max { 2 , γ } ∫ 0 ∞ ( D s ( ∇ w log f s ) + I s w , A ) d s ≤ max { 2 , γ } I . \begin{align*}
&2\int_0^\infty\mathcal I^{w,A}_s\,\mathrm ds-2\gamma\int_0^\infty\!\int f_s\sum_{i,j}(\partial_{y_i}\partial_{w_j}\log f_s)(\partial_{w_i}\partial_{w_j}\log f_s)\,\mathrm d\boldsymbol\mu_s\,\mathrm ds\\
\leq & 2\int_0^\infty\mathcal I^{w,A}_s\,\mathrm ds+\gamma\int_0^\infty\mathcal D_s(\nabla_w\log f_s)\,\mathrm ds\\
\leq & \max\{2,\gamma\}\int_0^\infty\bigl(\mathcal D_s(\nabla_w\log f_s)+\mathcal I^{w,A}_s\bigr)\,\mathrm ds\leq\max\{2,\gamma\}\,\mathcal I.
\end{align*} ≤ ≤ 2 ∫ 0 ∞ I s w , A d s − 2 γ ∫ 0 ∞ ∫ f s i , j ∑ ( ∂ y i ∂ w j log f s ) ( ∂ w i ∂ w j log f s ) d μ s d s 2 ∫ 0 ∞ I s w , A d s + γ ∫ 0 ∞ D s ( ∇ w log f s ) d s max { 2 , γ } ∫ 0 ∞ ( D s ( ∇ w log f s ) + I s w , A ) d s ≤ max { 2 , γ } I . For the force-defect derivative terms, the Cauchy–Schwarz inequality, (5.16) , and (5.13) bound the absolute value of their scale integral by
( ∫ 0 ∞ ∫ f s [ ∥ ∇ y r s ∥ H S 2 + ∥ ∇ w r s ∥ H S 2 ] d μ s d s ) 1 / 2 × ( ∫ 0 ∞ ∫ f s [ ∥ ∇ w 2 log f s ∥ H S 2 + ∥ ∇ y ∇ w log f s ∥ H S 2 ] d μ s d s ) 1 / 2 ≤ 2 η I E n t μ ( g t ) . \begin{align*}
&\left(\int_0^\infty\!\int f_s [\left\lVert \nabla_yr_s\right\rVert_{\mathrm{HS}}^2+\left\lVert \nabla_wr_s\right\rVert_{\mathrm{HS}}^2] \,\mathrm d\boldsymbol\mu_s\,\mathrm ds\right)^{1/2}\\
&\quad\times \left(\int_0^\infty\!\int f_s [\left\lVert \nabla_w^2\log f_s\right\rVert_{\mathrm{HS}}^2+\left\lVert \nabla_y\nabla_w\log f_s\right\rVert_{\mathrm{HS}}^2] \,\mathrm d\boldsymbol\mu_s\,\mathrm ds\right)^{1/2} \leq\sqrt2\eta\sqrt{\mathcal I\,\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)}.
\end{align*} ( ∫ 0 ∞ ∫ f s [ ∥ ∇ y r s ∥ HS 2 + ∥ ∇ w r s ∥ HS 2 ] d μ s d s ) 1/2 × ( ∫ 0 ∞ ∫ f s [ ∇ w 2 log f s HS 2 + ∥ ∇ y ∇ w log f s ∥ HS 2 ] d μ s d s ) 1/2 ≤ 2 η I Ent μ ( g t ) . For the friction cross term, by (4.23) , I s w ≤ I s w , A \mathcal I^w_s\leq\mathcal I^{w,A}_s I s w ≤ I s w , A (5.11) , we have
∣ J ( g t ) ∣ ≤ ( ∫ 0 ∞ I s y d s ) 1 / 2 ( ∫ 0 ∞ I s w d s ) 1 / 2 ≤ 2 E n t μ ( g t ) ( ∫ 0 ∞ I s w , A d s ) 1 / 2 ≤ 2 I E n t μ ( g t ) , \begin{align*}
\left\lvert \mathcal J(g_t)\right\rvert &\leq \Bigl(\int_0^\infty \mathcal I^y_s\,\mathrm ds\Bigr)^{1/2} \Bigl(\int_0^\infty \mathcal I^w_s\,\mathrm ds\Bigr)^{1/2}\\
&\leq\sqrt{2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)}\,\Bigl(\int_0^\infty \mathcal I^{w,A}_s\,\mathrm ds\Bigr)^{1/2} \leq\sqrt{2\mathcal I\,\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)},
\end{align*} ∣ J ( g t ) ∣ ≤ ( ∫ 0 ∞ I s y d s ) 1/2 ( ∫ 0 ∞ I s w d s ) 1/2 ≤ 2 Ent μ ( g t ) ( ∫ 0 ∞ I s w , A d s ) 1/2 ≤ 2 I Ent μ ( g t ) , implying − γ J ( g t ) ≤ 2 γ I E n t μ ( g t ) -\gamma\mathcal J(g_t)\leq\sqrt2\gamma\sqrt{\mathcal I\,\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)} − γ J ( g t ) ≤ 2 γ I Ent μ ( g t ) (5.7) gives
\begin{align*}
\frac\,\mathrm d{\,\mathrm dt}\mathcal J(g_t) \leq{}& -2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)+2\eta\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t) +\max\{2,\gamma\}\,\mathcal I+\frac{\eta}{\sqrt2}\mathcal I\\
&+\sqrt2\eta\sqrt{\mathcal I\,\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)} +\sqrt2\gamma\sqrt{\mathcal I\,\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)},
\end{align*}
which is (5.5) . The proof is complete. ◻
Kinetic derivative of the cross term
We prove Lemma 5.3 , which computes the kinetic derivative of the cross term at a fixed scale.
Proof of Lemma 5.3 . We fix s ≥ 0 s\geq0 s ≥ 0 h = log g t , s h=\log g_{t,s} h = log g t , s μ s \boldsymbol\mu_s μ s t t t A.2 justifies the differentiation under the integral and the integrations by parts below.
We first differentiate the integrand ∇ y g t , s ⋅ ∇ w g t , s / g t , s \nabla_yg_{t,s}\cdot\nabla_wg_{t,s}/g_{t,s} ∇ y g t , s ⋅ ∇ w g t , s / g t , s J s ( g t , s ) \mathcal J_s(g_{t,s}) J s ( g t , s )
∂ t J s ( g t , s ) = ∫ ∇ y ∂ t g t , s ⋅ ∇ w h + ∇ w ∂ t g t , s ⋅ ∇ y h − ∂ t g t , s ∇ y h ⋅ ∇ w h d μ s . \begin{equation}
\partial_t\mathcal J_s(g_{t,s})=\int \nabla_y\partial_tg_{t,s}\cdot\nabla_wh+\nabla_w\partial_tg_{t,s}\cdot\nabla_yh-\partial_tg_{t,s}\,\nabla_yh\cdot\nabla_wh \, \,\mathrm d\boldsymbol\mu_s. \tag*{(5.18)}
\end{equation} ∂ t J s ( g t , s ) = ∫ ∇ y ∂ t g t , s ⋅ ∇ w h + ∇ w ∂ t g t , s ⋅ ∇ y h − ∂ t g t , s ∇ y h ⋅ ∇ w h d μ s . (5.18) The right-hand side is linear in ∂ t g t , s \partial_tg_{t,s} ∂ t g t , s (4.30) . As noted before Lemma (?) , S w = − ∇ w ∗ ∇ w \mathcal S_w=-\nabla_w^*\nabla_w S w = − ∇ w ∗ ∇ w (4.29) and R t , s = g t , s r t , s R_{t,s}=g_{t,s}r_{t,s} R t , s = g t , s r t , s
γ S w g t , s = − ∇ w ∗ ( g t , s γ ∇ w h ) , − ∇ w ∗ R t , s = − ∇ w ∗ ( g t , s r t , s ) , \begin{equation*}
\gamma\mathcal S_wg_{t,s}=-\nabla_w^*\bigl(g_{t,s}\,\gamma\nabla_wh\bigr),\qquad
-\nabla_w^*R_{t,s}=-\nabla_w^*\bigl(g_{t,s}\,r_{t,s}\bigr),
\end{equation*} γ S w g t , s = − ∇ w ∗ ( g t , s γ ∇ w h ) , − ∇ w ∗ R t , s = − ∇ w ∗ ( g t , s r t , s ) , so we next treat the Hamiltonian term − A s g t , s -\mathcal A_sg_{t,s} − A s g t , s
Hamiltonian transport term. We substitute ∂ t g t , s = − A s g t , s \partial_tg_{t,s}=-\mathcal A_sg_{t,s} ∂ t g t , s = − A s g t , s (5.18) . Since A s \mathcal A_s A s
[ ∂ y i , A s ] = − ∑ j ( A + Q s ∇ 2 V s ( Q s y ) Q s ) i j ∂ w j , [ ∂ w i , A s ] = ∂ y i , \begin{equation*}
[\partial_{y_i},\mathcal A_s] =-\sum_j\bigl(A+Q_s\nabla^2V_s(Q_sy)Q_s\bigr)_{ij}\partial_{w_j}, \qquad [\partial_{w_i},\mathcal A_s]=\partial_{y_i},
\end{equation*} [ ∂ y i , A s ] = − j ∑ ( A + Q s ∇ 2 V s ( Q s y ) Q s ) ij ∂ w j , [ ∂ w i , A s ] = ∂ y i , which, along with ∇ g t , s = g t , s ∇ h \nabla g_{t,s}=g_{t,s}\nabla h ∇ g t , s = g t , s ∇ h A s ( ψ χ ) = ψ A s χ + χ A s ψ \mathcal A_s(\psi\chi)=\psi\,\mathcal A_s\chi+\chi\,\mathcal A_s\psi A s ( ψ χ ) = ψ A s χ + χ A s ψ
∫ ( ∇ y ∂ t g t , s ⋅ ∇ w h + ∇ w ∂ t g t , s ⋅ ∇ y h − ∂ t g t , s ∇ y h ⋅ ∇ w h ) = − ∫ A s ( g t , s ∇ y h ⋅ ∇ w h ) + ∫ g t , s [ ( ∇ w h ) ⊤ ( A + Q s ∇ 2 V s ( Q s y ) Q s ) ∇ w h − ∣ ∇ y h ∣ 2 ] . \begin{align*}
&\int\bigl(\nabla_y\partial_tg_{t,s}\cdot\nabla_wh+\nabla_w\partial_tg_{t,s}\cdot\nabla_yh-\partial_tg_{t,s}\,\nabla_yh\cdot\nabla_wh\bigr)\\
&\qquad=-\int\mathcal A_s\bigl(g_{t,s}\,\nabla_yh\cdot\nabla_wh\bigr)
+\int g_{t,s}\bigl[(\nabla_wh)^\top\bigl(A+Q_s\nabla^2V_s(Q_sy)Q_s\bigr)\nabla_wh-|\nabla_yh|^2\bigr].
\end{align*} ∫ ( ∇ y ∂ t g t , s ⋅ ∇ w h + ∇ w ∂ t g t , s ⋅ ∇ y h − ∂ t g t , s ∇ y h ⋅ ∇ w h ) = − ∫ A s ( g t , s ∇ y h ⋅ ∇ w h ) + ∫ g t , s [ ( ∇ w h ) ⊤ ( A + Q s ∇ 2 V s ( Q s y ) Q s ) ∇ w h − ∣ ∇ y h ∣ 2 ] . Since A s ∗ = − A s \mathcal A_s^*=-\mathcal A_s A s ∗ = − A s L 2 ( μ s ) L^2(\boldsymbol\mu_s) L 2 ( μ s ) A s 1 = 0 \mathcal A_s1=0 A s 1 = 0 (4.11) , the Hamiltonian transport term therefore gives the free Hamiltonian contribution and the smoothed interaction Hessian term in (5.7) :
− I s y ( g t , s ) + I s w , A ( g t , s ) + ∫ g t , s ( ∇ w h ) ⊤ Q s ∇ 2 V s ( Q s y ) Q s ∇ w h . \begin{equation*}
-\mathcal I^y_s(g_{t,s})+\mathcal I^{w,A}_s(g_{t,s})+\int g_{t,s}(\nabla_wh)^\top Q_s\nabla^2V_s(Q_sy)Q_s\nabla_wh.
\end{equation*} − I s y ( g t , s ) + I s w , A ( g t , s ) + ∫ g t , s ( ∇ w h ) ⊤ Q s ∇ 2 V s ( Q s y ) Q s ∇ w h . Momentum-divergence term. We consider ∂ t g t , s = − ∇ w ∗ ( g t , s r ) \partial_tg_{t,s}=-\nabla_w^*(g_{t,s}r) ∂ t g t , s = − ∇ w ∗ ( g t , s r ) r = γ ∇ w h r=\gamma\nabla_wh r = γ ∇ w h r = r t , s r=r_{t,s} r = r t , s ∇ w ∗ = − div w + w ⋅ \nabla_w^*=-\operatorname{div}_w+w\cdot ∇ w ∗ = − div w + w ⋅
∂ y i ∇ w ∗ ( g t , s r ) = ∇ w ∗ ( ∂ y i ( g t , s r ) ) , ∂ w i ∇ w ∗ ( g t , s r ) = ∇ w ∗ ( ∂ w i ( g t , s r ) ) + g t , s r i . \begin{equation*}
\partial_{y_i}\nabla_w^*(g_{t,s}r)=\nabla_w^*\bigl(\partial_{y_i}(g_{t,s}r)\bigr), \qquad \partial_{w_i}\nabla_w^*(g_{t,s}r)=\nabla_w^*\bigl(\partial_{w_i}(g_{t,s}r)\bigr)+g_{t,s}r_i.
\end{equation*} ∂ y i ∇ w ∗ ( g t , s r ) = ∇ w ∗ ( ∂ y i ( g t , s r ) ) , ∂ w i ∇ w ∗ ( g t , s r ) = ∇ w ∗ ( ∂ w i ( g t , s r ) ) + g t , s r i . Inserting these identities in (5.18) and integrating by parts in w w w
− ∫ ∑ i , j [ ∂ y i ( g t , s r j ) ∂ w i ∂ w j h + ∂ w i ( g t , s r j ) ∂ y i ∂ w j h ] − ∫ g t , s r ⋅ ∇ y h + ∫ g t , s r ⋅ ∇ w ( ∇ y h ⋅ ∇ w h ) . \begin{equation*}
-\int\sum_{i,j}\bigl[\partial_{y_i}(g_{t,s}r_j)\,\partial_{w_i}\partial_{w_j}h+\partial_{w_i}(g_{t,s}r_j)\,\partial_{y_i}\partial_{w_j}h\bigr]
-\int g_{t,s}\,r\cdot\nabla_yh+\int g_{t,s}\,r\cdot\nabla_w(\nabla_yh\cdot\nabla_wh).
\end{equation*} − ∫ i , j ∑ [ ∂ y i ( g t , s r j ) ∂ w i ∂ w j h + ∂ w i ( g t , s r j ) ∂ y i ∂ w j h ] − ∫ g t , s r ⋅ ∇ y h + ∫ g t , s r ⋅ ∇ w ( ∇ y h ⋅ ∇ w h ) . The extra term g t , s r i g_{t,s}r_i g t , s r i − ∫ g t , s r ⋅ ∇ y h -\int g_{t,s}\,r\cdot\nabla_yh − ∫ g t , s r ⋅ ∇ y h ∂ ( g t , s r j ) = g t , s ( ∂ r j + r j ∂ h ) \partial(g_{t,s}r_j)=g_{t,s}(\partial r_j+r_j\,\partial h) ∂ ( g t , s r j ) = g t , s ( ∂ r j + r j ∂ h ) ∂ = ∂ y i , ∂ w i \partial=\partial_{y_i},\partial_{w_i} ∂ = ∂ y i , ∂ w i r j ∂ h r_j\,\partial h r j ∂ h
r ⋅ ∇ w ( ∇ y h ⋅ ∇ w h ) = ∑ i , j r j [ ( ∂ y i h ) ∂ w i ∂ w j h + ( ∂ w i h ) ∂ y i ∂ w j h ] . \begin{equation*}
r\cdot\nabla_w(\nabla_yh\cdot\nabla_wh)=\sum_{i,j}r_j\bigl[(\partial_{y_i}h)\,\partial_{w_i}\partial_{w_j}h+(\partial_{w_i}h)\,\partial_{y_i}\partial_{w_j}h\bigr].
\end{equation*} r ⋅ ∇ w ( ∇ y h ⋅ ∇ w h ) = i , j ∑ r j [ ( ∂ y i h ) ∂ w i ∂ w j h + ( ∂ w i h ) ∂ y i ∂ w j h ] . The momentum divergence − ∇ w ∗ ( g t , s r ) -\nabla_w^*(g_{t,s}r) − ∇ w ∗ ( g t , s r )
− ∫ g t , s [ r ⋅ ∇ y h + ∑ i , j ( ∂ y i r j ) ∂ w i ∂ w j h + ∑ i , j ( ∂ w i r j ) ∂ y i ∂ w j h ] . \begin{equation*}
-\int g_{t,s}\Bigl[r\cdot\nabla_yh+\sum_{i,j}(\partial_{y_i}r_j)\,\partial_{w_i}\partial_{w_j}h+\sum_{i,j}(\partial_{w_i}r_j)\,\partial_{y_i}\partial_{w_j}h\Bigr].
\end{equation*} − ∫ g t , s [ r ⋅ ∇ y h + i , j ∑ ( ∂ y i r j ) ∂ w i ∂ w j h + i , j ∑ ( ∂ w i r j ) ∂ y i ∂ w j h ] . For r = r t , s r=r_{t,s} r = r t , s (5.7) . For r = γ ∇ w h r=\gamma\nabla_wh r = γ ∇ w h − γ J s ( g t , s ) -\gamma\mathcal J_s(g_{t,s}) − γ J s ( g t , s )
− 2 γ ∫ g t , s ∑ i , j ∂ y i ∂ w j h ∂ w i ∂ w j h , \begin{equation*}
-2\gamma\int g_{t,s}\sum_{i,j}\partial_{y_i}\partial_{w_j}h\,\partial_{w_i}\partial_{w_j}h,
\end{equation*} − 2 γ ∫ g t , s i , j ∑ ∂ y i ∂ w j h ∂ w i ∂ w j h , which are the two friction terms in (5.7) .
Combining the above computations completes the proof of (5.7) . ◻
Energy estimates along the Polchinski flow
We prove Lemma 5.4 and Proposition 5.5 . Throughout this subsection, f f f f s = P s f f_s=\mathsf P_sf f s = P s f (4.11) .
Both proofs rely on the same computational route. Specifically, for the proof of Lemma 5.4 , we note that in the momentum variable, the Polchinski generator is an Ornstein–Uhlenbeck generator independent of the interaction, so that ∂ s ∇ w f s = ( L s − 1 2 A ) ∇ w f s \partial_s\nabla_wf_s=(\mathsf L_s-\frac12A)\nabla_wf_s ∂ s ∇ w f s = ( L s − 2 1 A ) ∇ w f s (5.19) below. Consequently, the momentum Fisher information decreases from I p ( f ) \mathcal I_p(f) I p ( f ) ∇ y ∇ w log f s \nabla_y\nabla_w\log f_s ∇ y ∇ w log f s ∇ w 2 log f s \nabla_w^2\log f_s ∇ w 2 log f s
For the proof of Proposition 5.5 , (5.14) shows that the force defect R s R_s R s ∇ w f s \nabla_wf_s ∇ w f s Q s ∇ 2 V s ( Q s y ) Q s ∇ y f s Q_s\nabla^2V_s(Q_sy)Q_s\nabla_yf_s Q s ∇ 2 V s ( Q s y ) Q s ∇ y f s ∇ y f s \nabla_yf_s ∇ y f s R s R_s R s ( ∫ 0 ∞ b ( s ) ( I s y ) 1 / 2 d s ) 2 ≤ 2 η 2 E n t μ ( f ) \bigl(\int_0^\infty b(s)(\mathcal I^y_s)^{1/2}\,\mathrm ds\bigr)^2\leq2\eta^2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f) ( ∫ 0 ∞ b ( s ) ( I s y ) 1/2 d s ) 2 ≤ 2 η 2 Ent μ ( f )
Proof of Lemma 5.4 . Let ψ s = ∣ ∇ w f s ∣ 2 / f s \psi_s=|\nabla_wf_s|^2/f_s ψ s = ∣ ∇ w f s ∣ 2 / f s I s w = ∫ ψ s d μ s \mathcal I^w_s=\int\psi_s\,\mathrm d\boldsymbol\mu_s I s w = ∫ ψ s d μ s A.1 , ψ s \psi_s ψ s 4.1 , and (4.15) gives \frac\,\mathrm d{\,\mathrm ds}\mathcal I^w_s=\int(\partial_s-\mathsf L_s)\psi_s\,\mathrm d\boldsymbol\mu_s . Noting that the only term of L s \mathsf L_s L s w w w − 1 2 A w ⋅ ∇ w -\frac12Aw\cdot\nabla_w − 2 1 A w ⋅ ∇ w [ ∇ w , L s ] = − 1 2 A ∇ w [\nabla_w,\mathsf L_s]=-\frac12A\nabla_w [ ∇ w , L s ] = − 2 1 A ∇ w ∇ w \nabla_w ∇ w ∂ s f s = L s f s \partial_sf_s=\mathsf L_sf_s ∂ s f s = L s f s
∂ s ∇ w f s = L s ∇ w f s − 1 2 A ∇ w f s . \begin{equation}
\partial_s\nabla_wf_s=\mathsf L_s\nabla_wf_s-\frac12A\nabla_wf_s. \tag*{(5.19)}
\end{equation} ∂ s ∇ w f s = L s ∇ w f s − 2 1 A ∇ w f s . (5.19) By (5.19) above and ∂ s f s = L s f s \partial_sf_s=\mathsf L_sf_s ∂ s f s = L s f s (4.20) for L s \mathsf L_s L s
∂ s ψ s = 2 ∇ w log f s ⋅ ( L s ∇ w f s − 1 2 A ∇ w f s ) − ∣ ∇ w log f s ∣ 2 L s f s , L s ψ s = 2 ∇ w log f s ⋅ L s ∇ w f s − ∣ ∇ w log f s ∣ 2 L s f s + 2 f s ∑ i Γ s ( ∂ w i log f s ) . \begin{align*}
\partial_s\psi_s&=2\nabla_w\log f_s\cdot\Bigl(\mathsf L_s\nabla_wf_s-\frac12A\nabla_wf_s\Bigr)-|\nabla_w\log f_s|^2\,\mathsf L_sf_s,\\
\mathsf L_s\psi_s&=2\nabla_w\log f_s\cdot\mathsf L_s\nabla_wf_s-|\nabla_w\log f_s|^2\,\mathsf L_sf_s+2f_s\sum_i\Gamma_s(\partial_{w_i}\log f_s).
\end{align*} ∂ s ψ s L s ψ s = 2 ∇ w log f s ⋅ ( L s ∇ w f s − 2 1 A ∇ w f s ) − ∣ ∇ w log f s ∣ 2 L s f s , = 2 ∇ w log f s ⋅ L s ∇ w f s − ∣ ∇ w log f s ∣ 2 L s f s + 2 f s i ∑ Γ s ( ∂ w i log f s ) . The terms containing L s \mathsf L_s L s ( ∂ s − L s ) ψ s (\partial_s-\mathsf L_s)\psi_s ( ∂ s − L s ) ψ s
( ∂ s − L s ) ψ s = − f s ( ∇ w log f s ) ⊤ A ∇ w log f s − 2 f s ∑ i Γ s ( ∂ w i log f s ) . \begin{equation}
(\partial_s-\mathsf L_s)\psi_s=-f_s(\nabla_w\log f_s)^\top A\,\nabla_w\log f_s-2f_s\sum_i\Gamma_s(\partial_{w_i}\log f_s). \tag*{(5.20)}
\end{equation} ( ∂ s − L s ) ψ s = − f s ( ∇ w log f s ) ⊤ A ∇ w log f s − 2 f s i ∑ Γ s ( ∂ w i log f s ) . (5.20) Integrating against μ s \boldsymbol\mu_s μ s (5.8) gives (5.9) .
Since I s w , A ≥ I s w \mathcal I^{w,A}_s\geq\mathcal I^w_s I s w , A ≥ I s w A ⪰ I d A\succeq\mathrm{Id} A ⪰ Id \frac\,\mathrm d{\,\mathrm ds}\mathcal I^w_s\leq-\mathcal I^w_s . Then, using Gronwall’s inequality with I 0 w = I p ( f ) \mathcal I^w_0=\mathcal I_p(f) I 0 w = I p ( f ) (5.10) . Integrating (5.9) over [ 0 , S ] [0,S] [ 0 , S ]
I S w + ∫ 0 S ( D s ( ∇ w log f s ) + I s w , A ) d s = I p ( f ) , \begin{equation*}
\mathcal I^w_S+\int_0^S\bigl(\mathcal D_s(\nabla_w\log f_s)+\mathcal I^{w,A}_s\bigr)\,\mathrm ds=\mathcal I_p(f),
\end{equation*} I S w + ∫ 0 S ( D s ( ∇ w log f s ) + I s w , A ) d s = I p ( f ) , which implies (5.11) by I S w ≥ 0 \mathcal I^w_S\geq0 I S w ≥ 0 S → ∞ S\to\infty S → ∞ (5.12) follows from (5.8) , by A ⪰ I d A\succeq\mathrm{Id} A ⪰ Id s s s (5.11) with I s w , A ≥ 0 \mathcal I^{w,A}_s\geq0 I s w , A ≥ 0 (5.13) . ◻
Proof of Proposition 5.5 . We first derive the evolution (5.14) of R s R_s R s (4.27) , R s = Q s P s ( ∇ q f ) − ∇ y f s R_s=Q_s\mathsf P_s(\nabla_qf)-\nabla_yf_s R s = Q s P s ( ∇ q f ) − ∇ y f s
∂ s ( Q s P s ( ∇ q f ) ) = L s ( Q s P s ( ∇ q f ) ) − 1 2 A Q s P s ( ∇ q f ) , ∂ s ∇ y f s = L s ∇ y f s − 1 2 A ∇ y f s − Q s ∇ 2 V s ( Q s y ) Q s ∇ y f s . \begin{align}
\partial_s\bigl(Q_s\mathsf P_s(\nabla_qf)\bigr)&=\mathsf L_s\bigl(Q_s\mathsf P_s(\nabla_qf)\bigr)-\frac12A\,Q_s\mathsf P_s(\nabla_qf), \tag*{(5.21)} \\
\partial_s \nabla_y f_s & = \mathsf L_s\nabla_yf_s-\frac12A\nabla_yf_s-Q_s\nabla^2V_s(Q_sy)Q_s\nabla_yf_s. \tag*{(5.22)}
\end{align} ∂ s ( Q s P s ( ∇ q f ) ) ∂ s ∇ y f s = L s ( Q s P s ( ∇ q f ) ) − 2 1 A Q s P s ( ∇ q f ) , = L s ∇ y f s − 2 1 A ∇ y f s − Q s ∇ 2 V s ( Q s y ) Q s ∇ y f s . (5.21) (5.22) Subtracting the second equation from the first proves (5.14) . To obtain the first equation (5.21) , it suffices to apply Lemma 4.1 componentwise to ∇ q f ∈ C b ∞ ( R 2 d ) \nabla_qf\in C_b^\infty(\mathbb R^{2d}) ∇ q f ∈ C b ∞ ( R 2 d ) Q s Q_s Q s ( y , w ) (y,w) ( y , w ) L s \mathsf L_s L s \frac\,\mathrm d{\,\mathrm ds}Q_s=-\frac12AQ_s . For the second one (5.22) , noting that the only coefficient of L s \mathsf L_s L s y y y − ( 1 2 A y + Q s ∇ V s ( Q s y ) ) ⋅ ∇ y -\bigl(\frac12Ay+Q_s\nabla V_s(Q_sy)\bigr)\cdot\nabla_y − ( 2 1 A y + Q s ∇ V s ( Q s y ) ) ⋅ ∇ y
[ ∇ y , L s ] = − 1 2 A ∇ y − Q s ∇ 2 V s ( Q s y ) Q s ∇ y . \begin{equation*}
[\nabla_y,\mathsf L_s] = - \frac12A\nabla_y-Q_s\nabla^2V_s(Q_sy)Q_s\nabla_y.
\end{equation*} [ ∇ y , L s ] = − 2 1 A ∇ y − Q s ∇ 2 V s ( Q s y ) Q s ∇ y . Applying ∇ y \nabla_y ∇ y (4.13) , with ∇ y ∂ s f s = ∂ s ∇ y f s \nabla_y\partial_sf_s=\partial_s\nabla_yf_s ∇ y ∂ s f s = ∂ s ∇ y f s A.1 , gives (5.22) . The initial value R 0 = 0 R_0=0 R 0 = 0 (4.25) , since Q 0 = I d Q_0=\mathrm{Id} Q 0 = Id P 0 \mathsf P_0 P 0 V s ∣ s = 0 = V 0 V_s|_{s=0}=V_0 V s ∣ s = 0 = V 0
Next we derive the energy estimate. For convenience, we write
Y s : = ∫ ∣ R s ∣ 2 f s d μ s = ∫ f s ∣ r s ∣ 2 d μ s , σ s : = b ( s ) ( I s y ) 1 / 2 . \begin{equation}
Y_s := \int\frac{\left\lvert R_s\right\rvert^2}{f_s}\,\mathrm d\boldsymbol\mu_s = \int f_s|r_s|^2\,\mathrm d\boldsymbol\mu_s, \qquad \sigma_s := b(s)(\mathcal I^y_s)^{1/2}. \tag*{(5.23)}
\end{equation} Y s := ∫ f s ∣ R s ∣ 2 d μ s = ∫ f s ∣ r s ∣ 2 d μ s , σ s := b ( s ) ( I s y ) 1/2 . (5.23) Comparing (5.19) with (5.14) , a similar computation as in the proof of Lemma 5.4 for (5.20) , with R s R_s R s ∇ w f s \nabla_wf_s ∇ w f s
( ∂ s − L s ) ( ∣ R s ∣ 2 f s ) = − f s r s ⊤ A r s − 2 f s ∑ i Γ s ( r s , i ) + 2 r s ⊤ Q s ∇ 2 V s ( Q s y ) Q s ∇ y f s ⏟ source . \begin{equation*}
(\partial_s-\mathsf L_s)\Bigl(\frac{|R_s|^2}{f_s}\Bigr) =-f_sr_s^\top Ar_s-2f_s\sum_i\Gamma_s(r_{s,i})+\underbrace{2r_s^\top Q_s\nabla^2V_s(Q_sy)Q_s\nabla_yf_s}_{\text{source}}.
\end{equation*} ( ∂ s − L s ) ( f s ∣ R s ∣ 2 ) = − f s r s ⊤ A r s − 2 f s i ∑ Γ s ( r s , i ) + source 2 r s ⊤ Q s ∇ 2 V s ( Q s y ) Q s ∇ y f s . By Lemma A.1 , ∣ R s ∣ 2 / f s |R_s|^2/f_s ∣ R s ∣ 2 / f s 4.1 , and then integration against μ s \boldsymbol\mu_s μ s (4.15) ,
\begin{equation}
\frac\,\mathrm d{\,\mathrm ds}Y_s =-\mathcal D_s(r_s)-\int f_s r_s^\top Ar_s\,\mathrm d\boldsymbol\mu_s +2\int f_s r_s^\top Q_s\nabla^2V_s(Q_sy)Q_s\nabla_y\log f_s\,\mathrm d\boldsymbol\mu_s. \tag*{(5.24)}
\end{equation}
Since ∥ Q s ∇ 2 V s ( Q s y ) Q s ∥ o p ≤ b ( s ) \left\lVert Q_s\nabla^2V_s(Q_sy)Q_s\right\rVert_\mathrm{op}\leq b(s) Q s ∇ 2 V s ( Q s y ) Q s op ≤ b ( s ) y y y (5.24) , with σ s \sigma_s σ s (5.23) ,
2 ∫ f s r s ⊤ Q s ∇ 2 V s ( Q s y ) Q s ∇ y log f s d μ s ≤ 2 b ( s ) Y s ( I s y ) 1 / 2 = 2 Y s σ s . \begin{equation}
2\int f_s r_s^\top Q_s\nabla^2V_s(Q_sy)Q_s\nabla_y\log f_s\,\mathrm d\boldsymbol\mu_s\leq2b(s)\sqrt{Y_s}\,(\mathcal I^y_s)^{1/2}=2\sqrt{Y_s}\,\sigma_s. \tag*{(5.25)}
\end{equation} 2 ∫ f s r s ⊤ Q s ∇ 2 V s ( Q s y ) Q s ∇ y log f s d μ s ≤ 2 b ( s ) Y s ( I s y ) 1/2 = 2 Y s σ s . (5.25) It follows that
\begin{equation}
\frac\,\mathrm d{\,\mathrm ds}Y_s\leq2\sqrt{Y_s}\,\sigma_s\,, \qquad \sqrt{Y_s}\leq\int_0^s\sigma_{u}\,\mathrm du, \tag*{(5.26)}
\end{equation}
where the second inequality follows by integrating \frac\,\mathrm d{\,\mathrm ds}\sqrt{Y_s}\leq\sigma_s from Y 0 = 0 Y_0=0 Y 0 = 0 (5.24) over [ 0 , S ] [0,S] [ 0 , S ] Y 0 = 0 Y_0=0 Y 0 = 0 Y S ≥ 0 Y_S\geq0 Y S ≥ 0 (5.25) –(5.26) , we obtain
∫ 0 S [ D s ( r s ) + ∫ f s r s ⊤ A r s d μ s ] d s ≤ 2 ∫ 0 S σ s ( ∫ 0 s σ u d u ) d s = ( ∫ 0 S σ s d s ) 2 ≤ ( ∫ 0 S b ( s ) 2 d s ) ( ∫ 0 S I s y d s ) ≤ 2 η 2 E n t μ ( f ) . \begin{align}
\int_0^S \left[\mathcal D_s(r_s)+\int f_s r_s^\top Ar_s\,\mathrm d\boldsymbol\mu_s\right]\,\mathrm ds & \leq 2\int_0^S\sigma_s \left(\int_0^s\sigma_{u}\,\mathrm du\right)\,\mathrm ds =\left(\int_0^S\sigma_s\,\mathrm ds\right)^2 \notag \\
& \leq \left(\int_0^Sb(s)^2\,\mathrm ds\right) \left(\int_0^S\mathcal I^y_s\,\mathrm ds\right) \leq2\eta^2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f). \tag*{(5.27)}
\end{align} ∫ 0 S [ D s ( r s ) + ∫ f s r s ⊤ A r s d μ s ] d s ≤ 2 ∫ 0 S σ s ( ∫ 0 s σ u d u ) d s = ( ∫ 0 S σ s d s ) 2 ≤ ( ∫ 0 S b ( s ) 2 d s ) ( ∫ 0 S I s y d s ) ≤ 2 η 2 Ent μ ( f ) . (5.27) The last line follows from the Cauchy–Schwarz inequality in s s s (3.1) of η \eta η ∫ 0 ∞ I s y d s ≤ 2 E n t μ ( f ) \int_0^\infty\mathcal I^y_s\,\mathrm ds\leq2\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f) ∫ 0 ∞ I s y d s ≤ 2 Ent μ ( f ) (4.23) . Letting S → ∞ S\to\infty S → ∞ (5.15) .
Finally, since A ⪰ I d A\succeq\mathrm{Id} A ⪰ Id (5.8) gives
D s ( r s ) + ∫ f s r s ⊤ A r s d μ s ≥ ∫ f s [ ∥ ∇ y r s ∥ H S 2 + ∥ ∇ w r s ∥ H S 2 + ∣ r s ∣ 2 ] d μ s , \begin{equation*}
\mathcal D_s(r_s)+\int f_sr_s^\top Ar_s\,\mathrm d\boldsymbol\mu_s \geq\int f_s\bigl[\left\lVert \nabla_yr_s\right\rVert_{\mathrm{HS}}^2+\left\lVert \nabla_wr_s\right\rVert_{\mathrm{HS}}^2+|r_s|^2\bigr]\,\mathrm d\boldsymbol\mu_s,
\end{equation*} D s ( r s ) + ∫ f s r s ⊤ A r s d μ s ≥ ∫ f s [ ∥ ∇ y r s ∥ HS 2 + ∥ ∇ w r s ∥ HS 2 + ∣ r s ∣ 2 ] d μ s , and (5.16) follows. ◻
Application to the lattice sine–Gordon model
In this section, we prove Theorem 3.5 by rescaling to unit mass and bounding η \eta η BB ]. We then show that the mass order of the decay rate is optimal and compare it with the static logarithmic Sobolev inequality in [BB ].
Uniform bound on η \eta η
3.1 , we measure lengths and times in units of m − 1 m^{-1} m − 1 x ↦ m x x\mapsto mx x ↦ m x t ↦ m t t\mapsto mt t ↦ m t
τ = m t , q s c = m ε φ , p s c = ε ϖ . \begin{equation}
\tau=mt, \qquad q^{\mathrm{sc}}=m\varepsilon\varphi, \qquad p^{\mathrm{sc}}=\varepsilon\varpi. \tag*{(6.1)}
\end{equation} τ = m t , q sc = m εφ , p sc = ε ϖ . (6.1) The rescaling maps (2.18) to the same equation at unit mass, with parameters
( ε , L , z , m , γ ) ⟼ ( m ε , m L , z m − ( 2 − α ) , 1 , γ / m ) . \begin{equation}
(\varepsilon,L,z,m,\gamma) \longmapsto \bigl(m\varepsilon,\,mL,\,z\,m^{-(2-\alpha)},\,1,\,\gamma/m\bigr). \tag*{(6.2)}
\end{equation} ( ε , L , z , m , γ ) ⟼ ( m ε , m L , z m − ( 2 − α ) , 1 , γ / m ) . (6.2) Indeed, Δ ε = m 2 Δ m ε \Delta_\varepsilon=m^2\Delta_{m\varepsilon} Δ ε = m 2 Δ m ε
z ε 2 − α = ( z m − ( 2 − α ) ) ( m ε ) 2 − α , \begin{equation*}
z\varepsilon^{2-\alpha} =\bigl(z\,m^{-(2-\alpha)}\bigr)(m\varepsilon)^{2-\alpha},
\end{equation*} z ε 2 − α = ( z m − ( 2 − α ) ) ( m ε ) 2 − α , and Brownian scaling turns the noise into that of the rescaled friction. Let T \mathbf T T ( φ , ϖ ) ↦ ( q s c , p s c ) (\varphi,\varpi)\mapsto(q^{\mathrm{sc}},p^{\mathrm{sc}}) ( φ , ϖ ) ↦ ( q sc , p sc ) (6.1) . It sends a density f f f μ ε , L \boldsymbol\mu_{\varepsilon,L} μ ε , L f ∘ T − 1 f\circ\mathbf T^{-1} f ∘ T − 1 T # μ ε , L \mathbf T_\#\boldsymbol\mu_{\varepsilon,L} T # μ ε , L L 2 L^2 L 2
E n t T # μ ε , L ( f ∘ T − 1 ) = E n t μ ε , L ( f ) . \begin{equation}
\mathop{\mathrm{Ent}}_{\mathbf T_\#\boldsymbol\mu_{\varepsilon,L}}(f\circ\mathbf T^{-1}) =\mathop{\mathrm{Ent}}_{\boldsymbol\mu_{\varepsilon,L}}(f). \tag*{(6.3)}
\end{equation} Ent T # μ ε , L ( f ∘ T − 1 ) = Ent μ ε , L ( f ) . (6.3) As in § 3.2 , with a slight abuse of notation, we retain ε \varepsilon ε L L L z z z γ \gamma γ m = 1 m=1 m = 1 ( τ , q s c , p s c ) (\tau,q^{\mathrm{sc}},p^{\mathrm{sc}}) ( τ , q sc , p sc ) ( t , q , p ) (t,q,p) ( t , q , p )
d q t = p t d t , d p t = − ∇ U ( q t ) d t − γ p t d t + 2 γ d W t , \begin{equation}
\,\mathrm dq_t=p_t\,\mathrm dt, \qquad \,\mathrm dp_t=-\nabla U(q_t)\,\mathrm dt-\gamma p_t\,\mathrm dt +\sqrt{2\gamma}\,\mathrm dW_t, \tag*{(6.4)}
\end{equation} d q t = p t d t , d p t = − ∇ U ( q t ) d t − γ p t d t + 2 γ d W t , (6.4) with standard Euclidean Brownian motion and
U ( q ) = 1 2 ⟨ q , A q ⟩ + V 0 ( q ) , A = − Δ ε + I d ⪰ I d , \begin{equation}
U(q)=\frac12\left\langle q,Aq\right\rangle+V_0(q), \qquad A=-\Delta_\varepsilon+\mathrm{Id}\succeq\mathrm{Id}, \tag*{(6.5)}
\end{equation} U ( q ) = 2 1 ⟨ q , A q ⟩ + V 0 ( q ) , A = − Δ ε + Id ⪰ Id , (6.5) V 0 ( q ) = 2 z ε 2 − α ∑ x cos ( β ε q x ) . \begin{equation}
V_0(q)=2z\varepsilon^{2-\alpha} \sum_x\cos\left(\frac{\sqrt\beta}{\varepsilon}q_x\right). \tag*{(6.6)}
\end{equation} V 0 ( q ) = 2 z ε 2 − α x ∑ cos ( ε β q x ) . (6.6) The invariant measure is μ = ν ⊗ N ( 0 , I d ) \boldsymbol\mu=\nu\otimes\mathsf N(0,\mathrm{Id}) μ = ν ⊗ N ( 0 , Id ) ν ( d q ) ∝ e − U ( q ) d q \nu(\,\mathrm dq)\propto e^{-U(q)}\,\mathrm dq ν ( d q ) ∝ e − U ( q ) d q V 0 ∈ C b ∞ V_0\in C_b^\infty V 0 ∈ C b ∞ U U U (2.1) and the regularity results of Appendix § 7 apply.
η \eta η (6.6) , the function b b b η \eta η § 3.1 are
b ε , L ( s ) = sup q ∥ Q s ∇ 2 V s ( q ) Q s ∥ o p , η ε , L = ( ∫ 0 ∞ b ε , L ( s ) 2 d s ) 1 / 2 , \begin{equation}
b_{\varepsilon,L}(s) =\sup_q\left\lVert Q_s\nabla^2V_s(q)Q_s\right\rVert_{\mathrm{op}}, \qquad \eta_{\varepsilon,L} =\left(\int_0^\infty b_{\varepsilon,L}(s)^2\,\mathrm ds\right)^{1/2}, \tag*{(6.7)}
\end{equation} b ε , L ( s ) = q sup Q s ∇ 2 V s ( q ) Q s op , η ε , L = ( ∫ 0 ∞ b ε , L ( s ) 2 d s ) 1/2 , (6.7) which we will estimate for the sine–Gordon model. Let z c e ( β ) > 0 z_{\mathrm{ce}}(\beta)>0 z ce ( β ) > 0 δ β \delta_\beta δ β BB (3.5)], so that at unit mass the smallness condition there reads L ≥ 1 L\ge1 L ≥ 1 ∣ z ∣ ≤ z c e ( β ) |z|\le z_{\mathrm{ce}}(\beta) ∣ z ∣ ≤ z ce ( β ) η ε , L \eta_{\varepsilon,L} η ε , L ε \varepsilon ε L L L β < 4 π \beta<4\pi β < 4 π
Proposition 6.1 (Uniform bound on η \eta η 0 < β < 4 π 0<\beta<4\pi 0 < β < 4 π C β ′ > 0 C_\beta'>0 C β ′ > 0 β \beta β 0 < ε ≤ 1 0<\varepsilon\leq1 0 < ε ≤ 1 L ≥ 1 L\geq1 L ≥ 1 ∣ z ∣ ≤ z c e ( β ) \left\lvert z\right\rvert\leq z_{\mathrm{ce}}(\beta) ∣ z ∣ ≤ z ce ( β ) η ε , L ≤ C β ′ 1 − α ∣ z ∣ , α = β 4 π .
\eta_{\varepsilon,L} \leq \frac{C_\beta'}{\sqrt{1-\alpha}} \left\lvert z\right\rvert, \qquad \alpha=\frac{\beta}{4\pi}.
η ε , L ≤ 1 − α C β ′ ∣ z ∣ , α = 4 π β . ∣ z ∣ ≤ z ∗ ( β ) \left\lvert z\right\rvert\leq z_*(\beta) ∣ z ∣ ≤ z ∗ ( β ) η ε , L ≤ 1 / 2 \eta_{\varepsilon,L}\leq1/2 η ε , L ≤ 1/2 z ∗ ( β ) = min { z c e ( β ) , 1 − β / ( 4 π ) 2 C β ′ } .
z_*(\beta) =\min\left\{ z_{\mathrm{ce}}(\beta), \frac{\sqrt{1-\beta/(4\pi)}}{2C_\beta'} \right\}.
z ∗ ( β ) = min { z ce ( β ) , 2 C β ′ 1 − β / ( 4 π ) } .
Proof. For 0 < β < 4 π 0<\beta<4\pi 0 < β < 4 π ∣ z ∣ ≤ z c e ( β ) |z|\le z_{\mathrm{ce}}(\beta) ∣ z ∣ ≤ z ce ( β ) BB (3.53)]. They work on the unit lattice, with the field φ \varphi φ ε 2 A \varepsilon^2A ε 2 A t t t s = ε 2 t s=\varepsilon^2t s = ε 2 t Q t Q_t Q t Q s Q_s Q s q = ε φ q=\varepsilon\varphi q = εφ V t V_t V t ε 2 ∇ 2 V s \varepsilon^2\nabla^2V_s ε 2 ∇ 2 V s ϑ t 2 = e − s \vartheta_t^2=e^{-s} ϑ t 2 = e − s BB (3.15)] satisfies ε ℓ t = ℓ ( s ) \varepsilon\ell_t=\ell(s) ε ℓ t = ℓ ( s )
ℓ ( s ) = ( ε ∨ s ) ∧ 1 , \begin{equation*}
\ell(s)=(\varepsilon\vee\sqrt s)\wedge1,
\end{equation*} ℓ ( s ) = ( ε ∨ s ) ∧ 1 , and their microscopic coupling is O β ( ∣ z ∣ ε 2 − α ℓ t − α ) O_\beta(|z|\varepsilon^{2-\alpha}\ell_t^{-\alpha}) O β ( ∣ z ∣ ε 2 − α ℓ t − α ) BB (3.19)–(3.20)]. Therefore [BB (3.53)] gives
b ε , L ( s ) ≤ C β ∣ z ∣ ℓ ( s ) − α e − s , \begin{equation*}
b_{\varepsilon,L}(s)\leq C_\beta\left\lvert z\right\rvert\,\ell(s)^{-\alpha}e^{-s},
\end{equation*} b ε , L ( s ) ≤ C β ∣ z ∣ ℓ ( s ) − α e − s , with C β C_\beta C β β \beta β ℓ ( s ) \ell(s) ℓ ( s ) s s s [ 0 , ε 2 ] [0,\varepsilon^2] [ 0 , ε 2 ] [ ε 2 , 1 ] [\varepsilon^2,1] [ ε 2 , 1 ] [ 1 , ∞ ) [1,\infty) [ 1 , ∞ ) ℓ ( s ) \ell(s) ℓ ( s ) ε \varepsilon ε s \sqrt s s 1 1 1 e − s ≤ 1 e^{-s}\leq1 e − s ≤ 1 (3.9) . Since 0 < α < 1 0<\alpha<1 0 < α < 1 (3.9) gives
∫ 0 ∞ b ε , L ( s ) 2 d s ≤ C β 2 z 2 [ ε 2 − 2 α + 1 − ε 2 − 2 α 1 − α + e − 2 2 ] ≤ ( C β ′ ) 2 1 − α z 2 , \begin{align}
\int_0^\infty b_{\varepsilon,L}(s)^2\,\mathrm ds \leq C_\beta^2z^2 \left[ \varepsilon^{2-2\alpha} +\frac{1-\varepsilon^{2-2\alpha}}{1-\alpha} +\frac{e^{-2}}2 \right]
\leq \frac{(C_\beta')^2}{1-\alpha} z^2, \tag*{(6.10)}
\end{align} ∫ 0 ∞ b ε , L ( s ) 2 d s ≤ C β 2 z 2 [ ε 2 − 2 α + 1 − α 1 − ε 2 − 2 α + 2 e − 2 ] ≤ 1 − α ( C β ′ ) 2 z 2 , (6.10) with ( C β ′ ) 2 = C β 2 ( 2 + e − 2 / 2 ) (C_\beta')^2=C_\beta^2(2+e^{-2}/2) ( C β ′ ) 2 = C β 2 ( 2 + e − 2 /2 ) ε 2 − 2 α ≤ 1 ≤ ( 1 − α ) − 1 \varepsilon^{2-2\alpha}\leq1\leq(1-\alpha)^{-1} ε 2 − 2 α ≤ 1 ≤ ( 1 − α ) − 1 (6.8) . If ∣ z ∣ ≤ z ∗ ( β ) \left\lvert z\right\rvert\leq z_*(\beta) ∣ z ∣ ≤ z ∗ ( β ) (6.8) and (6.9) give η ε , L ≤ 1 / 2 \eta_{\varepsilon,L}\leq1/2 η ε , L ≤ 1/2
6.1 to prove Theorem 3.5 , and deduce the L 2 L^2 L 2
Proof of Theorem 3.5 . At each fixed lattice, the unit-mass model (6.4) satisfies the assumptions of Theorem 3.1 , and Proposition 6.1 gives η ε , L ≤ 1 / 2 \eta_{\varepsilon,L}\leq1/2 η ε , L ≤ 1/2 ∣ z ∣ ≤ z ∗ ( β ) \left\lvert z\right\rvert\leq z_*(\beta) ∣ z ∣ ≤ z ∗ ( β ) 3.1 with η ∗ = 1 / 2 \eta_*=1/2 η ∗ = 1/2 (3.2) with the constants (3.7) in unit-mass coordinates.
The map T \mathbf T T (2.18) at time t t t (6.4) at time τ = m t \tau=mt τ = m t T # μ ε , L = μ \mathbf T_\#\boldsymbol\mu_{\varepsilon,L}=\boldsymbol\mu T # μ ε , L = μ (6.3) at times 0 0 0 t t t e − c γ τ e^{-c_\gamma\tau} e − c γ τ (3.2) to e − c γ m t e^{-c_\gamma mt} e − c γ m t (3.6) at m = 1 m=1 m = 1 m m m
Corollary 6.2 (L 2 L^2 L 2 3.5 , every ψ ∈ L 2 ( μ ε , L ) \psi\in L^2(\boldsymbol\mu_{\varepsilon,L}) ψ ∈ L 2 ( μ ε , L ) E μ ε , L ψ = 0 \mathbb E_{\boldsymbol\mu_{\varepsilon,L}}\psi=0 E μ ε , L ψ = 0 ∥ P t ε , L ψ ∥ 2 ≤ C γ e − c γ t / 2 ∥ ψ ∥ 2 , t ≥ 0.
\left\lVert \mathcal P_t^{\varepsilon,L}\psi\right\rVert_2 \le\sqrt{C_\gamma}\,e^{-c_\gamma t/2}\left\lVert \psi\right\rVert_2, \qquad t\ge0.
P t ε , L ψ 2 ≤ C γ e − c γ t /2 ∥ ψ ∥ 2 , t ≥ 0.
Proof. Apply Corollary 3.3 in unit-mass coordinates and use that T \mathbf T T L 2 L^2 L 2
Optimality of the mass scaling
We keep m m m ε \varepsilon ε L L L z z z γ \gamma γ (2.18) . When γ \gamma γ m m m m − 1 m^{-1} m − 1 m m m 3.5 , which includes z = 0 z=0 z = 0 3.5 has the optimal order in the mass and the friction. We then compare the kinetic rate with the static Poincaré and logarithmic Sobolev constants of the position law and with the rate of the overdamped dynamics.
Let u u u ∥ u ∥ ε = 1 \left\lVert u\right\rVert_\varepsilon=1 ∥ u ∥ ε = 1 A ε u = m 2 u A_\varepsilon u=m^2u A ε u = m 2 u z = 0 z=0 z = 0 ( m φ u , ϖ u ) (m\varphi^u,\varpi^u) ( m φ u , ϖ u ) φ u = ⟨ φ , u ⟩ ε \varphi^{u}=\left\langle \varphi,u\right\rangle_\varepsilon φ u = ⟨ φ , u ⟩ ε ϖ u = ⟨ ϖ , u ⟩ ε \varpi^{u}=\left\langle \varpi,u\right\rangle_\varepsilon ϖ u = ⟨ ϖ , u ⟩ ε N ( 0 , I d 2 ) \mathsf N(0,\mathrm{Id}_2) N ( 0 , Id 2 )
M m , γ = ( 0 m − m − γ ) , λ ± ( M m , γ ) = − γ ± γ 2 − 4 m 2 2 . \begin{equation}
\mathbf M_{m,\gamma}=\begin{pmatrix}0&m\\-m&-\gamma\end{pmatrix},\qquad \lambda_\pm(\mathbf M_{m,\gamma})=\frac{-\gamma\pm\sqrt{\gamma^2-4m^2}}2. \tag*{(6.12)}
\end{equation} M m , γ = ( 0 − m m − γ ) , λ ± ( M m , γ ) = 2 − γ ± γ 2 − 4 m 2 . (6.12) Translating the equilibrium law by h ∈ R 2 h\in\mathbb R^2 h ∈ R 2 t t t e t M m , γ h e^{t\mathbf M_{m,\gamma}}h e t M m , γ h ∣ e t M m , γ h ∣ 2 / 2 |e^{t\mathbf M_{m,\gamma}}h|^2/2 ∣ e t M m , γ h ∣ 2 /2 E n t ( ϱ t ∣ μ ε , L ) ≤ C e − λ t E n t ( ϱ 0 ∣ μ ε , L ) \mathop{\mathrm{Ent}}(\boldsymbol\varrho_t\mid\boldsymbol\mu_{\varepsilon,L})\le Ce^{-\lambda t}\mathop{\mathrm{Ent}}(\boldsymbol\varrho_0\mid\boldsymbol\mu_{\varepsilon,L}) Ent ( ϱ t ∣ μ ε , L ) ≤ C e − λ t Ent ( ϱ 0 ∣ μ ε , L ) ∥ e t M m , γ ∥ o p 2 ≤ C e − λ t \left\lVert e^{t\mathbf M_{m,\gamma}}\right\rVert_\mathrm{op}^2\le Ce^{-\lambda t} e t M m , γ op 2 ≤ C e − λ t t ≥ 0 t\geq0 t ≥ 0
λ ≤ − 2 Re λ + ( M m , γ ) = γ − Re γ 2 − 4 m 2 . \begin{equation}
\lambda\le-2\operatorname{Re}\lambda_+(\mathbf M_{m,\gamma})=\gamma-\operatorname{Re}\sqrt{\gamma^2-4m^2}. \tag*{(6.13)}
\end{equation} λ ≤ − 2 Re λ + ( M m , γ ) = γ − Re γ 2 − 4 m 2 . (6.13) When γ \gamma γ m m m m m m
For large friction, (3.7) gives c γ ∼ 1 / ( 8 γ ) c_\gamma\sim1/(8\gamma) c γ ∼ 1/ ( 8 γ ) c γ c_\gamma c γ c γ m c_\gamma m c γ m m 2 / ( 8 γ ) m^2/(8\gamma) m 2 / ( 8 γ ) m 2 / γ m^2/\gamma m 2 / γ (6.13) .
We compare the kinetic rate with the static functional inequalities of the position law. Let ρ L S I \rho_{\mathrm{LSI}} ρ LSI ρ P I \rho_{\mathrm{PI}} ρ PI ν ε , L \nu_{\varepsilon,L} ν ε , L
E n t ν ε , L ( ψ 2 ) ≤ 2 ρ L S I D ε , L ( ψ ) , Var ν ε , L ( ψ ) ≤ 1 ρ P I D ε , L ( ψ ) , \begin{align}
\mathop{\mathrm{Ent}}_{\nu_{\varepsilon,L}}(\psi^2) &\le \frac{2}{\rho_{\mathrm{LSI}}}\mathcal D_{\varepsilon,L}(\psi),\notag\\
\operatorname{Var}_{\nu_{\varepsilon,L}}(\psi) &\le \frac{1}{\rho_{\mathrm{PI}}}\mathcal D_{\varepsilon,L}(\psi), \tag*{(6.14)}
\end{align} Ent ν ε , L ( ψ 2 ) Var ν ε , L ( ψ ) ≤ ρ LSI 2 D ε , L ( ψ ) , ≤ ρ PI 1 D ε , L ( ψ ) , (6.14) where
For unnormalized functions, E n t ν ( ψ 2 ) = ∫ ψ 2 log ( ψ 2 / E ν ψ 2 ) d ν \mathop{\mathrm{Ent}}_\nu(\psi^2)=\int\psi^2\log(\psi^2/\mathbb E_\nu\psi^2)\,\mathrm d\nu Ent ν ( ψ 2 ) = ∫ ψ 2 log ( ψ 2 / E ν ψ 2 ) d ν ρ L S I ≤ ρ P I \rho_{\mathrm{LSI}}\le\rho_{\mathrm{PI}} ρ LSI ≤ ρ PI z = 0 z=0 z = 0 ρ L S I = ρ P I = m 2 \rho_{\mathrm{LSI}}=\rho_{\mathrm{PI}}=m^2 ρ LSI = ρ PI = m 2
For small nonzero coupling, both constants remain of order m 2 m^2 m 2 BB ], and the upper bound follows from a variance lower bound for the spatial average of the field. Hence, for friction proportional to m m m 3.5 is of the order of the square root of either static constant.
Proposition 6.3 (Two-sided bounds on the static constants). For each 0 < β < 4 π 0<\beta<4\pi 0 < β < 4 π z s t a t ( β ) > 0 z_{\mathrm{stat}}(\beta)>0 z stat ( β ) > 0 0 < m ε ≤ 1 0<m\varepsilon\le1 0 < m ε ≤ 1 m L ≥ 1 mL\ge1 m L ≥ 1 ∣ z ∣ m − ( 2 − α ) ≤ min { z ∗ ( β ) , z s t a t ( β ) } ,
|z|\,m^{-(2-\alpha)}\le \min\{z_*(\beta),z_{\mathrm{stat}}(\beta)\},
∣ z ∣ m − ( 2 − α ) ≤ min { z ∗ ( β ) , z stat ( β )} , m 2 2 ≤ ρ L S I ≤ ρ P I ≤ v ∗ − 1 m 2 , v ∗ = ( 1 − e − 1 / 4 ) ( 2 − e 1 / 2 ) > 0.
\frac{m^2}{2}\le\rho_{\mathrm{LSI}}\le\rho_{\mathrm{PI}} \le v_*^{-1}m^2,\qquad v_*=(1-e^{-1/4})(2-e^{1/2})>0.
2 m 2 ≤ ρ LSI ≤ ρ PI ≤ v ∗ − 1 m 2 , v ∗ = ( 1 − e − 1/4 ) ( 2 − e 1/2 ) > 0.
Proof. The Hamiltonian (2.17) , the Dirichlet form (6.15) , and the normalization (6.14) are those of [BB (1.15)–(1.17) and (1.19)]. Since z ∗ ( β ) ≤ z c e ( β ) z_*(\beta)\le z_{\mathrm{ce}}(\beta) z ∗ ( β ) ≤ z ce ( β ) (6.16) satisfies the smallness condition of [BB Theorem 1.6], which is also [BB (3.5)], and [BB (1.20)] gives
ρ L S I ≥ m 2 − C β s t a t m α ∣ z ∣ = m 2 ( 1 − C β s t a t ∣ z ∣ m − ( 2 − α ) ) , \begin{equation*}
\rho_{\mathrm{LSI}}\geq m^2-C_\beta^{\mathrm{stat}}m^{\alpha}|z| =m^2\bigl(1-C_\beta^{\mathrm{stat}}|z|m^{-(2-\alpha)}\bigr),
\end{equation*} ρ LSI ≥ m 2 − C β stat m α ∣ z ∣ = m 2 ( 1 − C β stat ∣ z ∣ m − ( 2 − α ) ) , with C β s t a t C_\beta^{\mathrm{stat}} C β stat β \beta β z s t a t ( β ) = 1 / ( 2 C β s t a t ) z_{\mathrm{stat}}(\beta)=1/(2C_\beta^{\mathrm{stat}}) z stat ( β ) = 1/ ( 2 C β stat ) (6.17) .
For the upper bound, we bound from below the variance of the constant-mode coordinate φ u = ⟨ φ , u ⟩ ε \varphi^u=\left\langle \varphi ,u\right\rangle_\varepsilon φ u = ⟨ φ , u ⟩ ε ∥ ∇ ε φ u ∥ ε = ∥ u ∥ ε = 1 \left\lVert \nabla_\varepsilon\varphi^u\right\rVert_\varepsilon=\left\lVert u\right\rVert_\varepsilon=1 ∥ ∇ ε φ u ∥ ε = ∥ u ∥ ε = 1 q = m ε φ q=m\varepsilon\varphi q = m εφ (6.1) . The rescaled model has parameters ( m ε , m L , z m − ( 2 − α ) ) (m\varepsilon,mL,z\,m^{-(2-\alpha)}) ( m ε , m L , z m − ( 2 − α ) ) A = m − 2 A ε = − Δ m ε + I d A=m^{-2}A_\varepsilon=-\Delta_{m\varepsilon}+\mathrm{Id} A = m − 2 A ε = − Δ m ε + Id ν \nu ν V s V_s V s Q s Q_s Q s C s C_s C s b m ε , m L b_{m\varepsilon,mL} b m ε , m L η m ε , m L \eta_{m\varepsilon,mL} η m ε , m L u ^ \hat u u ^ A u ^ = u ^ A\hat u=\hat u A u ^ = u ^ φ u = m − 1 u ^ ⋅ q \varphi^u=m^{-1}\hat u\cdot q φ u = m − 1 u ^ ⋅ q (6.16) , Proposition 6.1 gives η m ε , m L ≤ 1 / 2 \eta_{m\varepsilon,mL}\le1/2 η m ε , m L ≤ 1/2 Var ν ( u ^ ⋅ q ) ≥ v ∗ \operatorname{Var}_\nu(\hat u\cdot q)\ge v_* Var ν ( u ^ ⋅ q ) ≥ v ∗
Specifically, we bound this variance through the posterior covariance at a suitable scale. For s > 0 s>0 s > 0 π s ( d q ∣ y ) \pi_s(\,\mathrm dq\mid y) π s ( d q ∣ y ) § 2.2 . Averaging the posteriors against ν s \nu_s ν s ν \nu ν (2.12) at Q s y Q_sy Q s y
e − V s ( Q s y ) = ∫ e − V 0 ( q ) N ( Q s y , C s ) ( d q ) . \begin{equation*}
e^{-V_s(Q_sy)}=\int e^{-V_0(q)}\,\mathsf N(Q_sy,C_s)(\,\mathrm dq).
\end{equation*} e − V s ( Q s y ) = ∫ e − V 0 ( q ) N ( Q s y , C s ) ( d q ) . Differentiating the logarithm of this Gaussian integral twice in its mean Q s y Q_sy Q s y
− ∇ 2 V s ( Q s y ) = C s − 1 Cov π s ( ⋅ ∣ y ) ( q ) C s − 1 − C s − 1 , \begin{equation*}
-\nabla^2V_s(Q_sy)=C_s^{-1}\operatorname{Cov}_{\pi_s(\cdot\mid y)}(q)\,C_s^{-1}-C_s^{-1},
\end{equation*} − ∇ 2 V s ( Q s y ) = C s − 1 Cov π s ( ⋅ ∣ y ) ( q ) C s − 1 − C s − 1 , that is (see also [BBD2024 Lemma 3.12, (3.61)]),
Cov π s ( ⋅ ∣ y ) ( q ) = C s − C s ∇ 2 V s ( Q s y ) C s . \begin{equation}
\operatorname{Cov}_{\pi_s(\cdot\mid y)}(q) =C_s-C_s\nabla^2V_s(Q_sy)C_s. \tag*{(6.18)}
\end{equation} Cov π s ( ⋅ ∣ y ) ( q ) = C s − C s ∇ 2 V s ( Q s y ) C s . (6.18) Since A u ^ = u ^ A\hat u=\hat u A u ^ = u ^ C s u ^ = ( 1 − e − s ) u ^ C_s\hat u=(1-e^{-s})\hat u C s u ^ = ( 1 − e − s ) u ^ Q s u ^ = e − s / 2 u ^ Q_s\hat u=e^{-s/2}\hat u Q s u ^ = e − s /2 u ^ b m ε , m L ( s ) b_{m\varepsilon,mL}(s) b m ε , m L ( s ) y y y
u ^ ⊤ ∇ 2 V s ( Q s y ) u ^ = e s u ^ ⊤ Q s ∇ 2 V s ( Q s y ) Q s u ^ ≤ e s b m ε , m L ( s ) . \begin{equation*}
\hat u^\top\nabla^2V_s(Q_sy)\hat u
=e^s\hat u^\top Q_s\nabla^2V_s(Q_sy)Q_s\hat u
\le e^sb_{m\varepsilon,mL}(s).
\end{equation*} u ^ ⊤ ∇ 2 V s ( Q s y ) u ^ = e s u ^ ⊤ Q s ∇ 2 V s ( Q s y ) Q s u ^ ≤ e s b m ε , m L ( s ) . The law of total variance and (6.18) give
Var ν ( u ^ ⋅ q ) ≥ E ν s [ u ^ ⊤ Cov π s ( ⋅ ∣ y ) ( q ) u ^ ] ≥ ( 1 − e − s ) − ( 1 − e − s ) 2 e s b m ε , m L ( s ) = ( 1 − e − s ) [ 1 − ( e s − 1 ) b m ε , m L ( s ) ] . \begin{equation}
\begin{aligned}
\operatorname{Var}_\nu(\hat u\cdot q)
&\geq\mathbb E_{\nu_s}\bigl[\hat u^\top\operatorname{Cov}_{\pi_s(\cdot\mid y)}(q)\,\hat u\bigr]\\
&\geq(1-e^{-s})-(1-e^{-s})^2e^sb_{m\varepsilon,mL}(s)\\
&=(1-e^{-s})\bigl[1-(e^s-1)b_{m\varepsilon,mL}(s)\bigr].
\end{aligned} \tag*{(6.19)}
\end{equation} Var ν ( u ^ ⋅ q ) ≥ E ν s [ u ^ ⊤ Cov π s ( ⋅ ∣ y ) ( q ) u ^ ] ≥ ( 1 − e − s ) − ( 1 − e − s ) 2 e s b m ε , m L ( s ) = ( 1 − e − s ) [ 1 − ( e s − 1 ) b m ε , m L ( s ) ] . (6.19) Since ∫ 1 / 4 1 / 2 b m ε , m L ( s ) 2 d s ≤ η m ε , m L 2 ≤ 1 / 4 \int_{1/4}^{1/2}b_{m\varepsilon,mL}(s)^2\,\mathrm ds\le\eta_{m\varepsilon,mL}^2\le1/4 ∫ 1/4 1/2 b m ε , m L ( s ) 2 d s ≤ η m ε , m L 2 ≤ 1/4 s ∈ [ 1 / 4 , 1 / 2 ] s\in[1/4,1/2] s ∈ [ 1/4 , 1/2 ] b m ε , m L ( s ) ≤ 1 b_{m\varepsilon,mL}(s)\le1 b m ε , m L ( s ) ≤ 1 (6.19) is at least v ∗ v_* v ∗ Var ν ( u ^ ⋅ q ) ≥ v ∗ \operatorname{Var}_\nu(\hat u\cdot q)\ge v_* Var ν ( u ^ ⋅ q ) ≥ v ∗
Thanks to the variance lower bound shown above, noting φ u = m − 1 u ^ ⋅ q \varphi^u=m^{-1}\hat u\cdot q φ u = m − 1 u ^ ⋅ q Var ν ε , L ( φ u ) ≥ v ∗ m − 2 \operatorname{Var}_{\nu_{\varepsilon,L}}(\varphi^u)\ge v_*m^{-2} Var ν ε , L ( φ u ) ≥ v ∗ m − 2 D ε , L ( φ u ) = 1 \mathcal D_{\varepsilon,L}(\varphi^u)=1 D ε , L ( φ u ) = 1 (6.14) gives
\begin{equation*}
\rho_{\mathrm{PI}}\leq\frac{\mathcal D_{\varepsilon,L}(\varphi^u)}{\operatorname{Var}_{\nu_{\varepsilon,L}}(\varphi^u)}\leq v_*^{-1}m^2.\qedhere
\end{equation*}
◻
Finally, we compare with the overdamped dynamics, whose rate depends on a choice of time normalization; we fix it by a mobility ω > 0 \omega>0 ω > 0 ω \omega ω
d φ t = − ω ∇ ε H ε , L ( φ t ) d t + 2 ω d W t ε . \begin{equation*}
\,\mathrm d\varphi_t=-\omega\nabla_\varepsilon H_{\varepsilon,L}(\varphi_t)\,\mathrm dt +\sqrt{2\omega}\,\mathrm dW_t^\varepsilon.
\end{equation*} d φ t = − ω ∇ ε H ε , L ( φ t ) d t + 2 ω d W t ε . Its logarithmic Sobolev estimate gives entropy decay with exponent 2 ω ρ L S I 2\omega\rho_{\mathrm{LSI}} 2 ω ρ LSI e − 2 ω m 2 t e^{-2\omega m^2t} e − 2 ω m 2 t ω = 1 \omega=1 ω = 1 m 2 m^2 m 2 m m m ω = 1 / γ \omega=1/\gamma ω = 1/ γ (2.18) ; with this choice, the overdamped exponent 2 m 2 / γ 2m^2/\gamma 2 m 2 / γ (6.13) and is of order m m m m m m
Regularity, scale limits, and approximation
In this appendix, we justify the differentiations and scale limits in § 4 –§ 5 , and extend the entropy estimate to finite-entropy initial laws by approximation. All regularity bounds are at a fixed cutoff and may depend on the model parameters.
Regularity and the scale limit
The kinetic stochastic flow and the Gaussian convolutions defining the coarse-grained densities provide the bounds needed for differentiation and integration by parts, and for the limit s → ∞ s\to\infty s → ∞ A A A V 0 V_0 V 0 γ > 0 \gamma>0 γ > 0 § 2.1 .
Lemma A.1 (Regularity along the Polchinski flow). The renormalized potential satisfies ∣ ∂ s [ V s ( Q s y ) ] ∣ ≤ C ( 1 + ∣ y ∣ ) |\partial_s[V_s(Q_sy)]|\leq C(1+|y|) ∣ ∂ s [ V s ( Q s y )] ∣ ≤ C ( 1 + ∣ y ∣ ) s ≥ 0 s\geq0 s ≥ 0 y ∈ R d y\in\mathbb R^d y ∈ R d f f f 0 < c − ≤ f ≤ c + 0<c_-\leq f\leq c_+ 0 < c − ≤ f ≤ c + c − ≤ f s ≤ c + c_-\leq f_s\leq c_+ c − ≤ f s ≤ c + k = 0 , 1 , … k=0,1,\ldots k = 0 , 1 , … s ≥ 0 s\geq0 s ≥ 0 ( y , w ) ∈ R 2 d (y,w)\in\mathbb R^{2d} ( y , w ) ∈ R 2 d ∣ ∇ ( y , w ) k f s ( y , w ) ∣ ≤ C k e − k s / 2 , ∣ ∂ s ∇ ( y , w ) k f s ( y , w ) ∣ ≤ C k e − k s / 2 ( 1 + ∣ ( y , w ) ∣ ) ,
|\nabla_{(y,w)}^kf_s(y,w)| \leq C_ke^{-ks/2},\qquad
|\partial_s\nabla_{(y,w)}^kf_s(y,w)| \leq C_ke^{-ks/2}(1+|(y,w)|),
∣ ∇ ( y , w ) k f s ( y , w ) ∣ ≤ C k e − k s /2 , ∣ ∂ s ∇ ( y , w ) k f s ( y , w ) ∣ ≤ C k e − k s /2 ( 1 + ∣ ( y , w ) ∣ ) , C k C_k C k f f f c − c_- c − ∥ ∇ ( q , p ) j f ∥ ∞ \|\nabla_{(q,p)}^jf\|_\infty ∥ ∇ ( q , p ) j f ∥ ∞ R s R_s R s r s r_s r s e − s / 2 e^{-s/2} e − s /2 f s log f s f_s\log f_s f s log f s ∣ ∇ w f s ∣ 2 / f s |\nabla_wf_s|^2/f_s ∣ ∇ w f s ∣ 2 / f s ∣ R s ∣ 2 / f s |R_s|^2/f_s ∣ R s ∣ 2 / f s 4.1 .
Proof. Conditional expectation preserves c − ≤ f s ≤ c + c_-\leq f_s\leq c_+ c − ≤ f s ≤ c + (4.7) , the denominator is at least e − sup V 0 e^{-\sup V_0} e − s u p V 0 ( y , w ) (y,w) ( y , w ) Q s Q_s Q s ∥ Q s ∥ o p ≤ e − s / 2 \left\lVert Q_s\right\rVert_\mathrm{op}\leq e^{-s/2} ∥ Q s ∥ op ≤ e − s /2 (7.1) .
(2.15) and the chain rule, with \frac\,\mathrm d{\,\mathrm ds}Q_s=-\frac12AQ_s , give
∂ s [ V s ( Q s y ) ] = 1 2 Q s 2 : ∇ 2 V s ( Q s y ) − 1 2 ∣ Q s ∇ V s ( Q s y ) ∣ 2 − 1 2 A y ⋅ Q s ∇ V s ( Q s y ) . \begin{equation}
\partial_s[V_s(Q_sy)] =\frac12Q_s^2:\nabla^2V_s(Q_sy) -\frac12\left\lvert Q_s\nabla V_s(Q_sy)\right\rvert^2 -\frac12Ay\cdot Q_s\nabla V_s(Q_sy). \tag*{(7.2)}
\end{equation} ∂ s [ V s ( Q s y )] = 2 1 Q s 2 : ∇ 2 V s ( Q s y ) − 2 1 ∣ Q s ∇ V s ( Q s y ) ∣ 2 − 2 1 A y ⋅ Q s ∇ V s ( Q s y ) . (7.2) Together with (4.13) , this provides the required scale derivatives. In particular, ∇ V s ( Q s y ) \nabla V_s(Q_sy) ∇ V s ( Q s y ) (2.14) of ∇ V 0 \nabla V_0 ∇ V 0 ∥ ∇ V s ∥ ∞ ≤ ∥ ∇ V 0 ∥ ∞ \left\lVert \nabla V_s\right\rVert_\infty\leq\left\lVert \nabla V_0\right\rVert_\infty ∥ ∇ V s ∥ ∞ ≤ ∥ ∇ V 0 ∥ ∞ ∣ Q s ∇ V s ( Q s y ) ∣ ≤ e − s / 2 ∥ ∇ V 0 ∥ ∞ |Q_s\nabla V_s(Q_sy)|\leq e^{-s/2}\left\lVert \nabla V_0\right\rVert_\infty ∣ Q s ∇ V s ( Q s y ) ∣ ≤ e − s /2 ∥ ∇ V 0 ∥ ∞
The second-order coefficients of L s \mathsf L_s L s ( y , w ) (y,w) ( y , w ) e − V s e^{-V_s} e − V s e − V s ≥ e − sup V 0 e^{-V_s}\geq e^{-\sup V_0} e − V s ≥ e − s u p V 0 V s V_s V s s s s ∣ ∇ y k [ Q s ∇ V s ( Q s y ) ] ∣ ≤ C k e − ( k + 1 ) s / 2 |\nabla_y^k[Q_s\nabla V_s(Q_sy)]|\leq C_ke^{-(k+1)s/2} ∣ ∇ y k [ Q s ∇ V s ( Q s y )] ∣ ≤ C k e − ( k + 1 ) s /2 ∇ ( y , w ) k \nabla_{(y,w)}^k ∇ ( y , w ) k ∂ s f s = L s f s \partial_sf_s=\mathsf L_sf_s ∂ s f s = L s f s (7.1) ; the linear factor comes from the drift. Equation (7.2) gives the stated bound on ∂ s [ V s ( Q s y ) ] \partial_s[V_s(Q_sy)] ∂ s [ V s ( Q s y )]
R s R_s R s Q s Q_s Q s f s ≥ c − f_s\geq c_- f s ≥ c −
∣ ∇ ( y , w ) k R s ∣ + ∣ ∇ ( y , w ) k r s ∣ ≤ C k e − ( k + 1 ) s / 2 , k ≥ 0 , \begin{equation*}
|\nabla_{(y,w)}^kR_s|+|\nabla_{(y,w)}^kr_s| \leq C_ke^{-(k+1)s/2},\qquad k\geq0,
\end{equation*} ∣ ∇ ( y , w ) k R s ∣ + ∣ ∇ ( y , w ) k r s ∣ ≤ C k e − ( k + 1 ) s /2 , k ≥ 0 , and (5.14) gives the corresponding scale derivatives.
Z 0 − 1 e − V s ( Q s y ) Z_0^{-1}e^{-V_s(Q_sy)} Z 0 − 1 e − V s ( Q s y ) C ( 1 + ∣ y ∣ ) C(1+|y|) C ( 1 + ∣ y ∣ ) f s ≥ c − f_s\geq c_- f s ≥ c − f s log f s f_s\log f_s f s log f s ∣ ∇ w f s ∣ 2 / f s |\nabla_wf_s|^2/f_s ∣ ∇ w f s ∣ 2 / f s ∣ R s ∣ 2 / f s |R_s|^2/f_s ∣ R s ∣ 2 / f s ( y , w ) (y,w) ( y , w ) ( y , w ) (y,w) ( y , w ) s ≥ 0 s\geq0 s ≥ 0 4.1 . The same estimates give polynomial majorants for the other integrands in § 4 –§ 5 and their scale derivatives, locally uniformly in s s s
Lemma A.2 (Regularity along the kinetic evolution). Let g 0 g_0 g 0 0 < c − ≤ g 0 ≤ c + 0<c_-\leq g_0\leq c_+ 0 < c − ≤ g 0 ≤ c + T < ∞ T<\infty T < ∞ g t = P t ∗ g 0 g_t=\mathcal P_t^*g_0 g t = P t ∗ g 0 0 ≤ t ≤ T 0\leq t\leq T 0 ≤ t ≤ T c − ≤ g t ≤ c + , sup 0 ≤ t ≤ T ∥ ∇ ( q , p ) k g t ∥ ∞ < ∞ ( k ≥ 1 ) ,
c_-\leq g_t\leq c_+, \qquad \sup_{0\leq t\leq T}\left\lVert \nabla_{(q,p)}^kg_t\right\rVert_\infty<\infty \quad(k\geq1),
c − ≤ g t ≤ c + , sup 0 ≤ t ≤ T ∇ ( q , p ) k g t ∞ < ∞ ( k ≥ 1 ) , A.1 applies to f = g t f=g_t f = g t t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ] k = 0 , 1 , … k=0,1,\ldots k = 0 , 1 , … sup 0 ≤ t ≤ T ∣ ∂ t ∇ ( q , p ) k g t ( q , p ) ∣ ≤ C T , k ( 1 + ∣ q ∣ + ∣ p ∣ ) , sup 0 ≤ t ≤ T ∣ ∂ t ∇ ( y , w ) k g t , s ( y , w ) ∣ ≤ C T , k e − k s / 2 ( 1 + ∣ ( y , w ) ∣ ) ,
\begin{gathered}
\sup_{0\leq t\leq T}|\partial_t\nabla_{(q,p)}^kg_t(q,p)| \leq C_{T,k}(1+|q|+|p|),\\
\sup_{0\leq t\leq T}|\partial_t\nabla_{(y,w)}^kg_{t,s}(y,w)| \leq C_{T,k}e^{-ks/2}(1+|(y,w)|),
\end{gathered}
0 ≤ t ≤ T sup ∣ ∂ t ∇ ( q , p ) k g t ( q , p ) ∣ ≤ C T , k ( 1 + ∣ q ∣ + ∣ p ∣ ) , 0 ≤ t ≤ T sup ∣ ∂ t ∇ ( y , w ) k g t , s ( y , w ) ∣ ≤ C T , k e − k s /2 ( 1 + ∣ ( y , w ) ∣ ) , R t , s R_{t,s} R t , s r t , s r_{t,s} r t , s e − s / 2 e^{-s/2} e − s /2
Proof. The reversed drift
( q , p ) ⟼ ( − p , ∇ U ( q ) − γ p ) \begin{equation*}
(q,p)\longmapsto(-p,\nabla U(q)-\gamma p)
\end{equation*} ( q , p ) ⟼ ( − p , ∇ U ( q ) − γ p ) is smooth and globally Lipschitz, with bounded derivatives of positive order. With momentum noise 2 γ d W t \sqrt{2\gamma}\,\mathrm dW_t 2 γ d W t P t ∗ \mathcal P_t^* P t ∗ c − ≤ g t ≤ c + c_-\leq g_t\leq c_+ c − ≤ g t ≤ c + [ 0 , T ] [0,T] [ 0 , T ] P t ∗ g 0 \mathcal P_t^*g_0 P t ∗ g 0 (7.3) . Since the drift grows at most linearly, the density equation, with the bounded spatial derivatives of g t g_t g t (7.4) . For the time derivative of g t , s g_{t,s} g t , s g t g_t g t ∂ t g t \partial_tg_t ∂ t g t (4.7) . Its derivatives have at most linear growth, while C s ⪯ A − 1 ⪯ I d C_s\preceq A^{-1}\preceq\mathrm{Id} C s ⪯ A − 1 ⪯ Id A C s ⪯ I d AC_s\preceq\mathrm{Id} A C s ⪯ Id ∂ t ∇ ( y , w ) k g t , s \partial_t\nabla_{(y,w)}^kg_{t,s} ∂ t ∇ ( y , w ) k g t , s R t , s R_{t,s} R t , s r t , s r_{t,s} r t , s
Lemma A.3 (Differentiation in kinetic time). Let g 0 g_0 g 0 T < ∞ T<\infty T < ∞ J ( g t ) \mathcal J(g_t) J ( g t ) t ↦ J ( g t ) t\mapsto\mathcal J(g_t) t ↦ J ( g t ) C 1 ( [ 0 , T ] ) C^1([0,T]) C 1 ([ 0 , T ])
\frac\,\mathrm d{\,\mathrm dt}\mathcal J(g_t) =\int_0^\infty\!\int \partial_tj_{t,s}\,\mathrm d\boldsymbol\mu_s\,\mathrm ds,
where j t , s = g t , s ∇ y log g t , s ⋅ ∇ w log g t , s j_{t,s}=g_{t,s}\nabla_y\log g_{t,s}\cdot\nabla_w\log g_{t,s} j t , s = g t , s ∇ y log g t , s ⋅ ∇ w log g t , s J s ( g t , s ) \mathcal J_s(g_{t,s}) J s ( g t , s ) t ↦ E n t μ ( g t ) t\mapsto\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t) t ↦ Ent μ ( g t ) C 1 ( [ 0 , T ] ) C^1([0,T]) C 1 ([ 0 , T ]) (2.7) .
Proof. Divide the first derivatives of g t , s g_{t,s} g t , s A.2 to obtain
∣ ∇ ( y , w ) log g t , s ∣ ≤ C T e − s / 2 , ∣ ∂ t ∇ ( y , w ) log g t , s ∣ ≤ C T e − s / 2 ( 1 + ∣ ( y , w ) ∣ ) . \begin{equation*}
|\nabla_{(y,w)}\log g_{t,s}|\leq C_Te^{-s/2},\qquad |\partial_t\nabla_{(y,w)}\log g_{t,s}| \leq C_Te^{-s/2}(1+|(y,w)|).
\end{equation*} ∣ ∇ ( y , w ) log g t , s ∣ ≤ C T e − s /2 , ∣ ∂ t ∇ ( y , w ) log g t , s ∣ ≤ C T e − s /2 ( 1 + ∣ ( y , w ) ∣ ) . By the product rule, the integrand j t , s j_{t,s} j t , s
sup 0 ≤ t ≤ T ( ∣ j t , s ( y , w ) ∣ + ∣ ∂ t j t , s ( y , w ) ∣ ) ≤ C T e − s ( 1 + ∣ ( y , w ) ∣ ) . \begin{equation}
\sup_{0\leq t\leq T} \bigl(|j_{t,s}(y,w)|+|\partial_tj_{t,s}(y,w)|\bigr) \leq C_Te^{-s}(1+|(y,w)|). \tag*{(7.6)}
\end{equation} 0 ≤ t ≤ T sup ( ∣ j t , s ( y , w ) ∣ + ∣ ∂ t j t , s ( y , w ) ∣ ) ≤ C T e − s ( 1 + ∣ ( y , w ) ∣ ) . (7.6) Moreover,
e − osc V 0 ≤ d μ s d N ( 0 , C ∞ ) ≤ e osc V 0 , s ≥ 0. \begin{equation*}
e^{-\operatorname{osc}V_0} \leq\frac{\,\mathrm d\boldsymbol\mu_s}{\,\mathrm d\mathsf N(0,\mathbf C_\infty)} \leq e^{\operatorname{osc}V_0},\qquad s\geq0.
\end{equation*} e − osc V 0 ≤ d N ( 0 , C ∞ ) d μ s ≤ e osc V 0 , s ≥ 0. Thus (7.6) provides a common integrable majorant on [ 0 , ∞ ) × R 2 d [0,\infty)\times\mathbb R^{2d} [ 0 , ∞ ) × R 2 d d s N ( 0 , C ∞ ) ( d y d w ) \,\mathrm ds\,\mathsf N(0,\mathbf C_\infty)(\,\mathrm dy\,\,\mathrm dw) d s N ( 0 , C ∞ ) ( d y d w ) (7.5) and continuity of its right-hand side.
The two-sided bounds on g t g_t g t ∂ t g t \partial_tg_t ∂ t g t (7.4) , and Gaussian moments justify differentiating E n t μ ( g t ) \mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t) Ent μ ( g t ) (2.7) . ◻
Lemma A.4 (Vanishing terminal entropy). For a regular density f f f f s → 1 f_s\to1 f s → 1 E n t μ s ( f s ) → 0 \mathop{\mathrm{Ent}}_{\boldsymbol\mu_s}(f_s)\to0 Ent μ s ( f s ) → 0 s → ∞ s\to\infty s → ∞
Proof. In (4.7) , Q s → 0 Q_s\to0 Q s → 0 C s → C ∞ \mathbf C_s\to\mathbf C_\infty C s → C ∞ f f f V 0 V_0 V 0 E N ( 0 , C ∞ ) [ e − V 0 f ] = Z 0 E μ f = Z 0 \mathbb E_{\mathsf N(0,\mathbf C_\infty)}[e^{-V_0}f]=Z_0\mathbb E_{\boldsymbol\mu}f=Z_0 E N ( 0 , C ∞ ) [ e − V 0 f ] = Z 0 E μ f = Z 0 e − V s ( Q s y ) → Z 0 e^{-V_s(Q_sy)}\to Z_0 e − V s ( Q s y ) → Z 0 f s → 1 f_s\to1 f s → 1 inf f ≤ f s ≤ sup f \inf f\leq f_s\leq\sup f inf f ≤ f s ≤ sup f inf V 0 ≤ V s ≤ sup V 0 \inf V_0\leq V_s\leq\sup V_0 inf V 0 ≤ V s ≤ sup V 0 μ s \boldsymbol\mu_s μ s N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ )
E n t μ s ( f s ) = Z 0 − 1 ∫ f s log f s e − V s ( Q s y ) N ( 0 , C ∞ ) ( d y d w ) , \begin{equation*}
\mathop{\mathrm{Ent}}_{\boldsymbol\mu_s}(f_s)
=Z_0^{-1}\int f_s\log f_s\,e^{-V_s(Q_sy)}\,\mathsf N(0,\mathbf C_\infty)(\,\mathrm dy\,\,\mathrm dw),
\end{equation*} Ent μ s ( f s ) = Z 0 − 1 ∫ f s log f s e − V s ( Q s y ) N ( 0 , C ∞ ) ( d y d w ) , the integrand is therefore uniformly bounded and tends to zero pointwise. Dominated convergence proves the claim. ◻
Approximation of finite-entropy densities
We complete the proof of Theorem 3.1 by approximating the initial density in L 1 L^1 L 1
Completion of the proof of Theorem 3.1 . Let g 0 g_0 g 0 μ \boldsymbol\mu μ g 0 g_0 g 0 n n n n − 1 n^{-1} n − 1 ( q , p ) (q,p) ( q , p ) n n n g 0 ( n ) g_0^{(n)} g 0 ( n ) g 0 g_0 g 0 L 1 ( μ ) L^1(\boldsymbol\mu) L 1 ( μ ) P t ∗ \mathcal P_t^* P t ∗ L 1 ( μ ) L^1(\boldsymbol\mu) L 1 ( μ ) g t ( n ) = P t ∗ g 0 ( n ) → g t g_t^{(n)}=\mathcal P_t^*g_0^{(n)}\to g_t g t ( n ) = P t ∗ g 0 ( n ) → g t L 1 ( μ ) L^1(\boldsymbol\mu) L 1 ( μ )
\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t)
\leq\liminf_{n\to\infty}\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_t^{(n)})
\leq C_{\gamma,\eta_*}e^{-c_{\gamma,\eta_*}t}\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(g_0).\qedhere
◻
Identification with the formulation of Bauerschmidt and Bodineau
The change of variables Q s \mathbf Q_s Q s § 4 with the formulation of Bauerschmidt and Bodineau [BB ] and the survey [BBD2024 ]. We give the correspondence for the measure, semigroup, generator, and entropy decomposition.
The pushforward ( Q s ) # μ s = Z 0 − 1 e − V s ( q ) N ( 0 , Q s C ∞ Q s ) ( d q d p ) (\mathbf Q_s)_\#\boldsymbol\mu_s=Z_0^{-1}e^{-V_s(q)}\mathsf N(0,\mathbf Q_s\mathbf C_\infty\mathbf Q_s)(\,\mathrm dq\,\,\mathrm dp) ( Q s ) # μ s = Z 0 − 1 e − V s ( q ) N ( 0 , Q s C ∞ Q s ) ( d q d p ) (4.2) in [BB (1.5)] and [BBD2024 Definition 3.2]. The function f s ∘ Q s − 1 f_s\circ\mathbf Q_s^{-1} f s ∘ Q s − 1 f f f BBD2024 (3.12)] from scale 0 0 0 s s s ( 0 , s ) (0,s) ( 0 , s ) C s \mathbf C_s C s ( 0 , t ) (0,t) ( 0 , t ) ( y , w ) (y,w) ( y , w ) N ( 0 , C ∞ ) \mathsf N(0,\mathbf C_\infty) N ( 0 , C ∞ )
Applying the chain rule to (4.13) , with \frac\,\mathrm d{\,\mathrm ds}\mathbf Q_s^{-1}=\frac12\operatorname{diag}(A,A)\mathbf Q_s^{-1} , gives the generator of f s ∘ Q s − 1 f_s\circ\mathbf Q_s^{-1} f s ∘ Q s − 1 ( q , p ) (q,p) ( q , p )
\begin{equation*}
\frac12\Bigl(\frac\,\mathrm d{\,\mathrm ds}\mathbf C_s\Bigr):\nabla_{(q,p)}^2-(Q_s^2\nabla V_s)\cdot\nabla_q.
\end{equation*}
The Ornstein–Uhlenbeck drifts cancel against the derivative of Q s − 1 \mathbf Q_s^{-1} Q s − 1 BB (2.8)], with \bigl(\frac\,\mathrm d{\,\mathrm ds}\mathbf C_s\bigr):\nabla_{(q,p)}^2=Q_s^2:\nabla_q^2+AQ_s^2:\nabla_p^2 the Laplacian associated with \frac\,\mathrm d{\,\mathrm ds}\mathbf C_s . Since Q s Q_s Q s A A A (4.2) gives
\begin{equation}
\mathbf Q_s^{-1}\Bigl(\frac\,\mathrm d{\,\mathrm ds}\mathbf C_s\Bigr)\mathbf Q_s^{-1}
=\begin{pmatrix}\mathrm{Id}&0\\0&A\end{pmatrix}. \tag*{(8.1)}
\end{equation}
For smooth ψ \psi ψ ∇ ( q , p ) ( ψ ∘ Q s − 1 ) = Q s − 1 ( ∇ ( y , w ) ψ ) ∘ Q s − 1 \nabla_{(q,p)}(\psi\circ\mathbf Q_s^{-1})=\mathbf Q_s^{-1}(\nabla_{(y,w)}\psi)\circ\mathbf Q_s^{-1} ∇ ( q , p ) ( ψ ∘ Q s − 1 ) = Q s − 1 ( ∇ ( y , w ) ψ ) ∘ Q s − 1 (8.1) gives
\begin{equation}
\frac12\Bigl\langle\nabla_{(q,p)}(\psi\circ\mathbf Q_s^{-1}), \Bigl(\frac\,\mathrm d{\,\mathrm ds}\mathbf C_s\Bigr)\nabla_{(q,p)}(\psi\circ\mathbf Q_s^{-1})\Bigr\rangle\circ \mathbf Q_s =\frac12|\nabla_y\psi|^2+\frac12(\nabla_w\psi)^\top A\,\nabla_w\psi=\Gamma_s(\psi). \tag*{(8.2)}
\end{equation}
The left-hand side is the carré du champ of the generator of Bauerschmidt and Bodineau, composed with Q s \mathbf Q_s Q s E n t ( Q s ) # μ s ( f s ∘ Q s − 1 ) = E n t μ s ( f s ) \mathop{\mathrm{Ent}}_{(\mathbf Q_s)_\#\boldsymbol\mu_s}(f_s\circ\mathbf Q_s^{-1})=\mathop{\mathrm{Ent}}_{\boldsymbol\mu_s}(f_s) Ent ( Q s ) # μ s ( f s ∘ Q s − 1 ) = Ent μ s ( f s ) ψ = log f s \psi=\log f_s ψ = log f s (8.2) transforms the identity [BB (2.16)–(2.17)],
\begin{equation*}
\mathop{\mathrm{Ent}}_{\boldsymbol\mu}(f)=\frac12\int_0^\infty\!\!\int (f_s\circ\mathbf Q_s^{-1})\Bigl\langle\nabla_{(q,p)}\log(f_s\circ\mathbf Q_s^{-1}), \Bigl(\frac\,\mathrm d{\,\mathrm ds}\mathbf C_s\Bigr)\nabla_{(q,p)}\log(f_s\circ\mathbf Q_s^{-1})\Bigr\rangle \,\mathrm d(\mathbf Q_s)_\#\boldsymbol\mu_s\,\mathrm ds,
\end{equation*}
into (4.23) .
