The convergence and uniqueness of a discrete-time nonlinear Markov chain

TL;DR

This paper proves the convergence and uniqueness of a broad class of discrete-time nonlinear Markov chains, with applications in discrete differential geometry, including curvature flows and Laplacian equations, and introduces a new curvature concept.

Contribution

It establishes convergence and uniqueness results for nonlinear Markov chains, generalizes curvature flows, and defines a consistent nonlinear Ollivier Ricci curvature.

Findings

Discrete-time Ollivier Ricci curvature flow converges to a constant curvature metric.

Laplacian separation flow converges to the constant Laplacian solution.

Results apply to nonlinear Dirichlet forms and Perron-Frobenius theory.

Abstract

In this paper, we prove the convergence and uniqueness of a general discrete-time nonlinear Markov chain with specific conditions. The results have important applications in discrete differential geometry. First, we prove the discrete-time Ollivier Ricci curvature flow $d_{n+1}:=(1-\alpha\kappa_{d_{n}})d_{n}$ converges to a constant curvature metric on a finite weighted graph. As shown in \cite[Theorem 5.1]{M23}, a Laplacian separation principle holds on a locally finite graph with nonnegative Ollivier curvature. We further prove that the Laplacian separation flow converges to the constant Laplacian solution and generalize the result to nonlinear $p$-Laplace operators. Moreover, our results can also be applied to study the long-time behavior in the nonlinear Dirichlet forms theory and nonlinear Perron-Frobenius theory. Finally, we define the Ollivier Ricci curvature of the nonlinear Markov chain which is consistent with the classical Ollivier Ricci curvature, sectional curvature \cite{CMS24}, coarse Ricci curvature on hypergraphs \cite{IKTU21} and the modified Ollivier Ricci curvature for $p$-Laplace. We also establish the convergence results for the nonlinear Markov chain with nonnegative Ollivier Ricci curvature.