Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Ruowei Li, Florentin Münch

TL;DR

The paper develops a general framework for discrete-time nonlinear Markov chains $P:\Omega\to\Omega$ under monotonicity, non-expansion, and accumulation-point conditions, proving convergence and, in many cases, uniqueness of long-time limits. It introduces a nonlinear Ollivier Ricci curvature for such chains and studies curvature-driven flows, including a discrete-time Ollivier Ricci curvature flow and a Laplacian separation flow, proving convergence to constant-curvature configurations and establishing gradient estimates via curvature bounds. The authors further connect these results to nonlinear Dirichlet forms and nonlinear Perron–Frobenius theory, and they define a consistent nonlinear curvature $\mathrm{Ric}_{r}(P,d)$ that recovers classical notions in special cases. Together, these results provide a rigorous foundation for analyzing long-time behavior and curvature-driven dynamics on graphs in the nonlinear setting, with implications for discrete differential geometry and nonlinear spectral 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 $d_{n+1}:=(1-ακ_{d_{n}})d_{n}$ 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.

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

TL;DR

The paper develops a general framework for discrete-time nonlinear Markov chains under monotonicity, non-expansion, and accumulation-point conditions, proving convergence and, in many cases, uniqueness of long-time limits. It introduces a nonlinear Ollivier Ricci curvature for such chains and studies curvature-driven flows, including a discrete-time Ollivier Ricci curvature flow and a Laplacian separation flow, proving convergence to constant-curvature configurations and establishing gradient estimates via curvature bounds. The authors further connect these results to nonlinear Dirichlet forms and nonlinear Perron–Frobenius theory, and they define a consistent nonlinear curvature that recovers classical notions in special cases. Together, these results provide a rigorous foundation for analyzing long-time behavior and curvature-driven dynamics on graphs in the nonlinear setting, with implications for discrete differential geometry and nonlinear spectral 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 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 -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 -Laplace. We also establish the convergence results for the nonlinear Markov chain with nonnegative Ollivier Ricci curvature.
Paper Structure (15 sections, 16 theorems, 54 equations, 1 table)

This paper contains 15 sections, 16 theorems, 54 equations, 1 table.

Table of Contents

  1. Introduction
  2. Convergence and uniqueness of nonlinear Markov chains
  3. Proofs of main theorems.
  4. An example of non-convergence.
  5. Basic facts of graphs
  6. Weighted graphs.
  7. Discrete Ricci curvature.
  8. Nonlinear Laplace and resolvent operators.
  9. Applications
  10. The convergence of Ollivier Ricci curvature flow.
  11. The gradient estimate for resolvents of nonlinear Laplace.
  12. The convergence of Laplacian separation flow.
  13. The nonlinear Dirichlet form.
  14. The nonlinear Perron-Frobenius theory.
  15. The Ollivier Ricci curvature of nonlinear Markov chains

Key Result

Theorem 1

Let $f\in \mathbb{R}^{N}$. If a discrete-time nonlinear Markov chain $P:\mathbb{R}^{N}\to\mathbb{R}^{N}$ satisfies (1) monotonicity, (2) strict monotonicity of corresponding components, (5) non-expansion, (9) $P^{n}f$ has a finite accumulation point $g\in \mathbb{R}^{N}$, then $Pg=g$ and $P^{n}f\to

Theorems & Definitions (30)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Theorem 4
  • Remark 3
  • Theorem 5
  • Theorem 6
  • ...and 20 more