Fokker-Planck equations on discrete infinite graphs

TL;DR

This paper explores the gradient flow structure and long-term behavior of Fokker-Planck equations on infinite graphs, establishing existence, convergence, and inequalities in an infinite-dimensional setting.

Contribution

It introduces a novel Hilbert manifold framework for FPE on infinite graphs and proves global existence, exponential convergence, and a Talagrand-type inequality.

Findings

Solutions converge exponentially to Gibbs distribution.

Established a Talagrand-type inequality in the infinite graph setting.

Compared the manifold metric with classical Wasserstein distances.

Abstract

We study the gradient flow structure and long-time behavior of Fokker-Planck equations (FPE) on infinite graphs, along with a Talagrand-type inequality in this setting. We begin by constructing an infinite-dimensional Hilbert manifold structure, extending the approach of [S. N. Chow, W. Huang, Y. Li, H. M. Zhou, Arch. Ration. Mech. Anal., 203, 969-1008 (2012)] through a novel classification method to establish injectivity of the map from quotient space to tangent space and employing functional analysis techniques to prove surjectivity. Using a combination of the relative energy method, approximation techniques, and continuity arguments, we establish the global existence and asymptotic convergence of solutions to the infinite-dimensional ODE system associated with the FPE. Specifically, we demonstrate that the FPE admits a gradient flow structure, with solutions converging exponentially to the unique Gibbs distribution. Furthermore, we prove a local Talagrand-type inequality and compare the Hilbert manifold metric induced by our framework with classical Wasserstein distances.