Logarithmic Sobolev inequalities for non-equilibrium steady states

TL;DR

This paper develops two novel methods to establish log-Sobolev inequalities for diffusion processes with non-explicit invariant measures and non-positive curvature regions, expanding tools for non-equilibrium steady states analysis.

Contribution

It introduces two approaches combining stochastic control, Wasserstein contraction, and hypercontractivity to derive log-Sobolev inequalities in complex non-equilibrium settings.

Findings

First method uses perturbation and stochastic control techniques.

Second method combines Wasserstein contraction with hypercontractivity.

Applicable to diffusion processes with non-explicit invariant measures.

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 Poincar\'e inequality in some non-equilibrium cases, with Wang's hypercontractivity results [Potential Anal., 53(3):1123-1144, 2020].