Fokker-Planck equations on discrete infinite graphs
Jose A. Carrillo, Xinyu Wang
TL;DR
The paper develops a rigorous gradient-flow framework for Fokker-Planck equations on infinite graphs by embedding probability densities into an infinite-dimensional Hilbert manifold and equipping it with a graph-induced metric. The authors prove global existence, exponential convergence to a unique Gibbs distribution, and a local Talagrand-type inequality, all within this discrete, infinite setting. A novel classification method ensures a bijection from a quotient space to the manifold's tangent space, enabling a well-defined gradient flow and long-time analysis via relative energy functions. Additionally, the work connects the Hilbert-manifold distance to the classical Wasserstein distance, offering insights into the geometric structure of entropy-driven dynamics on infinite graphs with sender-network weights.
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.