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Hypoelliptic entropy dissipation for stochastic differential equations

Qi Feng, Wuchen Li

TL;DR

This work develops a unified framework for analyzing convergence of general degenerate and non-reversible SDEs by introducing a weighted relative Fisher information as a Lyapunov functional. A structure condition yields a modified Hessian tensor $\mathfrak{R}$, and an information Bochner-type formula relates dissipation to a positive curvature bound $\kappa$, enabling exponential decay of $\mathcal{I}_{a,z}(p\|\pi)$ and, consequently, of the KL divergence and $L^1$ distance to the invariant density $\pi$. The theory accommodates hypoelliptic diffusion and non-gradient drifts without requiring iterative symmetry between diffusion components, and it delivers explicit convergence rates in key examples. Two detailed instantiations are provided: (i) underdamped Langevin dynamics with variable diffusion, and (ii) a chain of three oscillators with nearest-neighbor coupling, each illustrating how the Hessian structure controls the convergence rate. The results have potential implications for sampling, MCMC, and molecular dynamics where degenerate noise and irreversibility are common.

Abstract

We study the convergence analysis for general degenerate and non-reversible stochastic differential equations (SDEs). We apply the Lyapunov method to analyze the Fokker-Planck equation, in which the Lyapunov functional is chosen as a weighted relative Fisher information functional. We derive a structure condition and formulate the Lyapunov constant explicitly. We prove the exponential convergence result for the probability density function towards its invariant distribution in the $L_1$ distance. Two examples are presented: underdamped Langevin dynamics with variable diffusion matrices and three oscillator chain models with nearest-neighbor couplings.

Hypoelliptic entropy dissipation for stochastic differential equations

TL;DR

This work develops a unified framework for analyzing convergence of general degenerate and non-reversible SDEs by introducing a weighted relative Fisher information as a Lyapunov functional. A structure condition yields a modified Hessian tensor , and an information Bochner-type formula relates dissipation to a positive curvature bound , enabling exponential decay of and, consequently, of the KL divergence and distance to the invariant density . The theory accommodates hypoelliptic diffusion and non-gradient drifts without requiring iterative symmetry between diffusion components, and it delivers explicit convergence rates in key examples. Two detailed instantiations are provided: (i) underdamped Langevin dynamics with variable diffusion, and (ii) a chain of three oscillators with nearest-neighbor coupling, each illustrating how the Hessian structure controls the convergence rate. The results have potential implications for sampling, MCMC, and molecular dynamics where degenerate noise and irreversibility are common.

Abstract

We study the convergence analysis for general degenerate and non-reversible stochastic differential equations (SDEs). We apply the Lyapunov method to analyze the Fokker-Planck equation, in which the Lyapunov functional is chosen as a weighted relative Fisher information functional. We derive a structure condition and formulate the Lyapunov constant explicitly. We prove the exponential convergence result for the probability density function towards its invariant distribution in the distance. Two examples are presented: underdamped Langevin dynamics with variable diffusion matrices and three oscillator chain models with nearest-neighbor couplings.

Paper Structure

This paper contains 17 sections, 22 theorems, 239 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Settings
  4. Structure condition and Hessian matrix
  5. Entropy dissipation
  6. Example of Hessian matrices
  7. Example I: underdamped Langevin dynamics
  8. Example II: $3$-oscillators with nearest-neighbor couplings
  9. Proof
  10. Outline of main results
  11. Information Gamma calculus
  12. Proof of Step 1.
  13. Proof of Step 2.
  14. Derivation of $\widetilde{\Gamma}_{2}(f,f)$
  15. Derivation of $\widetilde{\Gamma}_{2,z}(f,f)$ calculus
  16. ...and 2 more sections

Key Result

Proposition 1

Denote a vector field $\gamma\colon \mathbb{R}^{n+m}\rightarrow\mathbb{R}^{n+m}$ satisfying Then equation FPE is equivalent to the following equation: In addition,

Figures (2)

  • Figure 1: Illustration of convergence rates $\kappa$ with different choices of $\beta$. Left: $\beta=0$; Right $\beta=0.1$.
  • Figure 2: Illustration of convergence rates $\kappa$ with different choices of $\beta$. Left: $\beta=0$; Right: $\beta=0.6$.

Theorems & Definitions (60)

  • Proposition 1: Decomposition
  • proof
  • Remark 1: Structure conditions
  • Definition 1: Hessian matrix
  • Remark 2
  • Remark 3
  • Remark 4: Connections with Bakry--Émery conditions
  • Theorem 1: Weighted Fisher information dissipation
  • Corollary 2: Functionals decay
  • Remark 5
  • ...and 50 more