Uniform log-Sobolev inequalities for mean field particles with flat-convex energy

TL;DR

This paper establishes uniform log-Sobolev inequalities for mean field particle systems with flat-convex energy, demonstrating long-term measure concentration and building on previous defective inequalities by tightening them with a uniform Poincaré inequality.

Contribution

It proves a uniform log-Sobolev inequality for flat-convex energy functionals in mean field systems, extending prior work by incorporating a uniform Poincaré inequality.

Findings

Proves uniform log-Sobolev inequalities for the system.

Shows measure concentration in the long-term behavior.

Builds on previous defective inequalities by tightening them.

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.