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

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