Uniform log-Sobolev inequalities for mean field particles with flat-convex energy
Abstract
The purpose of this short note is to demonstrate uniform logarithmic Sobolev inequalities for the mean field gradient particle systems associated to an energy functional that is convex in the flat sense. A defective log-Sobolev inequality was already established implicitly in a previous joint work with F. Chen and Z. Ren [arXiv:2212.03050 [math.PR]]. It remains only to tighten it by a uniform Poincar\'e inequality, which we prove by the method in a recent work of Guillin, W. Liu, L. Wu and C. Zhang [Ann. Appl. Probab., 32(3):1590-1614, 2022]. As an application, we show that the particle system exhibits the concentration of measure phenomenon in the long time.
Keywords
Cite
@article{arxiv.2408.03283,
title = {Uniform log-Sobolev inequalities for mean field particles with flat-convex energy},
author = {Songbo Wang},
journal= {arXiv preprint arXiv:2408.03283},
year = {2024}
}
Comments
17 pages; added comments on long-time concentration; fixed typos, especially one in the title