Limits of relative entropies associated with weakly interacting particle systems

TL;DR

This paper investigates the behavior of scaled relative entropies in weakly interacting N-particle Markov systems, establishing their convergence and exploring their role as Lyapunov functions in ergodic processes.

Contribution

It provides new results on the convergence of scaled relative entropies in weakly interacting particle systems, linking entropy limits to Lyapunov functions for Markov processes.

Findings

Convergence of scaled relative entropies established in various settings

Relative entropy acts as a Lyapunov function for ergodic Markov processes

Analysis supports understanding of nonlinear Markov process stability

Abstract

The limits of scaled relative entropies between probability distributions associated with N-particle weakly interacting Markov processes are considered. The convergence of such scaled relative entropies is established in various settings. The analysis is motivated by the role relative entropy plays as a Lyapunov function for the (linear) Kolmogorov forward equation associated with an ergodic Markov process, and Lyapunov function properties of these scaling limits with respect to nonlinear finite-state Markov processes are studied in the companion paper [6].