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arXiv · 2309.07858

Logarithmic Sobolev inequalities for non-equilibrium steady states

Pierre Monmarché·Songbo Wang
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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].

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Pierre Monmarché, Songbo Wang. 2023-09-14. Logarithmic Sobolev inequalities for non-equilibrium steady states. https://doi.org/10.1007/s11118-025-10211-6

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