A sufficient condition for the quasipotential to be the rate function of the invariant measure of countable-state mean-field interacting particle systems

TL;DR

This paper investigates when the Freidlin-Wentzell quasipotential accurately describes the large deviation behavior of invariant measures in countable-state mean-field particle systems, highlighting conditions where it fails or holds.

Contribution

It provides sufficient conditions ensuring the quasipotential is the correct rate function, clarifying its limitations through counterexamples and finite horizon considerations.

Findings

Counterexamples show quasipotential can differ from the rate function.

Barriers in finite horizon can prevent crossing, affecting the quasipotential.

Sufficient conditions are identified where quasipotential equals the rate function.

Abstract

This paper considers the family of invariant measures of Markovian mean-field interacting particle systems on a countably infinite state space and studies its large deviation asymptotics. The Freidlin-Wentzell quasipotential is the usual candidate rate function for the sequence of invariant measures indexed by the number of particles. The paper provides two counterexamples where the quasipotential is not the rate function. The quasipotential arises from finite horizon considerations. However there are certain barriers that cannot be surmounted easily in any finite time horizon, but these barriers can be crossed in the stationary regime. Consequently, the quasipotential is infinite at some points where the rate function is finite. After highlighting this phenomenon, the paper studies some sufficient conditions on a class of interacting particle systems under which one can continue to assert that the Freidlin-Wentzell quasipotential is indeed the rate function.