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Logarithmic Sobolev inequalities for non-equilibrium steady states

Pierre Monmarché, Songbo Wang

TL;DR

This work tackles the problem of proving log-Sobolev inequalities for invariant measures of diffusion processes whose densities are not explicit and whose curvature is not uniformly positive. It develops two complementary strategies: a perturbation approach around an explicit reference measure μ0, using Holley–Stroock and Aida–Shigekawa with a stochastic-control/HJB representation (addressing both elliptic and kinetic cases); and a defective LSI route based on Wasserstein-2 contraction and Wang’s hypercontractivity to combine Poincaré with a defective LSI, yielding constructive or explicit constants under suitable contraction conditions. The elliptic and kinetic perturbation results provide explicit LSI constants under clear contraction or Lipschitz growth assumptions, while the defective-LSI framework allows extending convergence results to non-reversible or degenerate settings through hypercontractivity and Poincaré properties. Overall, the paper offers a principled pathway to certify LSI for non-equilibrium steady states (NESS) in high-dimensional, non-explicit settings, including McKean–Vlasov and non-equilibrium models, with an emphasis on obtaining quantifiable constants.

Abstract

We consider two methods to establish log-Sobolev inequalities for the invariant measure of a diffusion process when its density is not explicit and the curvature is not positive everywhere. In the first approach, based on the Holley-Stroock and Aida-Shigekawa perturbation arguments [J. Stat. Phys., 46(5-6):1159-1194, 1987, J. Funct. Anal., 126(2):448-475, 1994], the control on the (non-explicit) perturbation is obtained by stochastic control methods, following the comparison technique introduced by Conforti [Ann. Appl. Probab., 33(6A):4608-4644, 2023]. The second method combines the Wasserstein-2 contraction method, used in [Ann. Henri Lebesgue, 6:941-973, 2023] to prove a Poincaré inequality in some non-equilibrium cases, with Wang's hypercontractivity results [Potential Anal., 53(3):1123-1144, 2020].

Logarithmic Sobolev inequalities for non-equilibrium steady states

TL;DR

This work tackles the problem of proving log-Sobolev inequalities for invariant measures of diffusion processes whose densities are not explicit and whose curvature is not uniformly positive. It develops two complementary strategies: a perturbation approach around an explicit reference measure μ0, using Holley–Stroock and Aida–Shigekawa with a stochastic-control/HJB representation (addressing both elliptic and kinetic cases); and a defective LSI route based on Wasserstein-2 contraction and Wang’s hypercontractivity to combine Poincaré with a defective LSI, yielding constructive or explicit constants under suitable contraction conditions. The elliptic and kinetic perturbation results provide explicit LSI constants under clear contraction or Lipschitz growth assumptions, while the defective-LSI framework allows extending convergence results to non-reversible or degenerate settings through hypercontractivity and Poincaré properties. Overall, the paper offers a principled pathway to certify LSI for non-equilibrium steady states (NESS) in high-dimensional, non-explicit settings, including McKean–Vlasov and non-equilibrium models, with an emphasis on obtaining quantifiable constants.

Abstract

We consider two methods to establish log-Sobolev inequalities for the invariant measure of a diffusion process when its density is not explicit and the curvature is not positive everywhere. In the first approach, based on the Holley-Stroock and Aida-Shigekawa perturbation arguments [J. Stat. Phys., 46(5-6):1159-1194, 1987, J. Funct. Anal., 126(2):448-475, 1994], the control on the (non-explicit) perturbation is obtained by stochastic control methods, following the comparison technique introduced by Conforti [Ann. Appl. Probab., 33(6A):4608-4644, 2023]. The second method combines the Wasserstein-2 contraction method, used in [Ann. Henri Lebesgue, 6:941-973, 2023] to prove a Poincaré inequality in some non-equilibrium cases, with Wang's hypercontractivity results [Potential Anal., 53(3):1123-1144, 2020].
Paper Structure (12 sections, 13 theorems, 142 equations)

This paper contains 12 sections, 13 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Perturbation approach: the elliptic case
  4. Perturbation approach: the kinetic case
  5. Defective LSI approach
  6. Proofs
  7. The elliptic case
  8. The kinetic case
  9. Defective log-Sobolev inequality
  10. From Hypercontractivity to defective LSI
  11. Hypercontractivity in the elliptic case
  12. Reflection coupling for kinetic diffusions

Key Result

Theorem 1

Assume that $\Sigma = \mathrm{Id}$ and $b=b_0 + b_1$ for some $b_0$, $b_1\in\mathcal{C}^1(\mathbb R^d,\mathbb R^d)$ with bounded derivatives such that the generator $\mathcal{L}_0 = b_0\cdot \nabla + \Delta$ admits a unique $C^2$ and positive invariant probability density $\mu_0$ satisfying an LSI Finally, assume that the law of $Z_t$ solving eq:EDS converges weakly for all initial condition as

Theorems & Definitions (30)

  • Theorem 1
  • Example 1
  • Example 2
  • Example 3
  • Theorem 2
  • Remark 1
  • Lemma 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • ...and 20 more