Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes

TL;DR

This paper investigates the local stability of nonlinear ODEs derived from finite-state Markov processes, introducing PDE-based Lyapunov functions and expanding the class of Gibbs systems with practical examples.

Contribution

It develops PDE-based methods to construct Lyapunov functions for nonlinear Markov process limits, extending stability analysis to a broader class of locally Gibbs systems.

Findings

PDE-based Lyapunov functions effectively analyze stability.

Explicit Lyapunov functions constructed for various locally Gibbs models.

Broader class of systems with stability properties identified.

Abstract

The focus of this work is on local stability of a class of nonlinear ordinary differential equations (ODE) that describe limits of empirical measures associated with finite-state weakly interacting N-particle systems. Local Lyapunov functions are identified for several classes of such ODE, including those associated with systems with slow adaptation and Gibbs systems. Using results from [5] and large deviations heuristics, a partial differential equation (PDE) associated with the nonlinear ODE is introduced and it is shown that positive definite subsolutions of this PDE serve as local Lyapunov functions for the ODE. This PDE characterization is used to construct explicit Lyapunov functions for a broad class of models called locally Gibbs systems. This class of models is significantly larger than the family of Gibbs systems and several examples of such systems are presented, including models with nearest neighbor jumps and models with simultaneous jumps that arise in applications.