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Gradient flow structure, well-posedness and asymptotic behavior of Fokker-Planck equation on locally finite graphs

Cong Wang

TL;DR

This work extends the gradient-flow interpretation of the Fokker-Planck equation from finite graphs to infinite, locally uniformly finite graphs by constructing a 2-Wasserstein-type metric on the probability density space. The FP equation is shown to be the gradient flow of the free energy $\\mathcal{F}(\\bm{\\rho}) = \sum_i \\pi_i\\Psi_i\\rho_i + \\sum_i \\pi_i\\rho_i\\log\\rho_i$ on the infinite-dimensional manifold $(\\mathcal{P}_0^*(G),\\mathcal{W}_2)$, with the Gibbs density $\\rho_i^* = \frac{1}{K}e^{-\\Psi_i}$ serving as the global attractor. The authors establish the existence and uniqueness of global solutions within $\\mathcal{P}_0^*(G)$ and prove convergence to the Gibbs state in $\\ell^{r}(V,\\bm{\\pi})$ norms for $r\\in[2,\\infty]$, providing the first rigorous Wasserstein-type metric framework and FP analysis on infinite graphs. These results lay groundwork for understanding diffusion and equilibration on large or infinite networks under a variational, metric-measure-theoretic lens.

Abstract

This paper investigates the gradient flow structure, well-posedness, and asymptotic behavior of the Fokker-Planck equation defined on locally uniformly finite graphs, which is highly non-trivial compared with the finite case. We first construct a 2-Wasserstein-type metric and gradient flow equation in the probability density space associated with the underlying graphs. Then, we prove the global existence of solution to the Fokker-Planck equation using a novel approach that differs significantly from the methods applied in the finite case. We also demonstrate that the solution converges to the Gibbs distribution in the $\ell^{r}(V,\bmπ)$ norm with $r\in [2,\infty]$, by using the indicator set partitioning method. To the best of our knowledge, this work seems the first result on the study of Wasserstein-type metrics and the Fokker-Planck equation in probability density spaces defined on infinite graphs.

Gradient flow structure, well-posedness and asymptotic behavior of Fokker-Planck equation on locally finite graphs

TL;DR

This work extends the gradient-flow interpretation of the Fokker-Planck equation from finite graphs to infinite, locally uniformly finite graphs by constructing a 2-Wasserstein-type metric on the probability density space. The FP equation is shown to be the gradient flow of the free energy on the infinite-dimensional manifold , with the Gibbs density serving as the global attractor. The authors establish the existence and uniqueness of global solutions within and prove convergence to the Gibbs state in norms for , providing the first rigorous Wasserstein-type metric framework and FP analysis on infinite graphs. These results lay groundwork for understanding diffusion and equilibration on large or infinite networks under a variational, metric-measure-theoretic lens.

Abstract

This paper investigates the gradient flow structure, well-posedness, and asymptotic behavior of the Fokker-Planck equation defined on locally uniformly finite graphs, which is highly non-trivial compared with the finite case. We first construct a 2-Wasserstein-type metric and gradient flow equation in the probability density space associated with the underlying graphs. Then, we prove the global existence of solution to the Fokker-Planck equation using a novel approach that differs significantly from the methods applied in the finite case. We also demonstrate that the solution converges to the Gibbs distribution in the norm with , by using the indicator set partitioning method. To the best of our knowledge, this work seems the first result on the study of Wasserstein-type metrics and the Fokker-Planck equation in probability density spaces defined on infinite graphs.

Paper Structure

This paper contains 6 sections, 11 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wasserstein type distance on locally uniformly finite graphs
  4. Wsserstein type distance I
  5. Wasserstein type distance II
  6. Fokker-Planck equation on locally uniformly finite graphs: proof of Theorem \ref{['thm:globalexistence']}

Key Result

Theorem 1.1

Let $G = (V, E, \bm{\pi})$ be a connected, locally uniformly finite and stochastically complete graph. Then, the following statements hold: (1) If $\bm{\rho}^0\in\mathcal{P}_0^*(G)$, the Fokker-Planck equation defined on the graph $G$ is a gradient flow on the infinite dimensional Riemannian manifol

Theorems & Definitions (22)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Definition 3.1
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 12 more