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

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Operations research and management science Limit theorems Special processes

Published online by Cambridge University Press:  21 March 2024

Sarath Yasodharan  and
Rajesh Sundaresan
Sarath Yasodharan*
Affiliation:
Brown University
Rajesh Sundaresan*
Affiliation:
Indian Institute of Science
*
*Postal address: Division of Applied Mathematics, 182 George Street, Providence, RI 02912, USA. Email address: sarath@alum.iisc.ac.in
**Postal address: Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru 560012, India. Email address: rajeshs@iisc.ac.in
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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.

Keywords

Mean-field interactioninvariant measurelarge deviationsstatic large deviationFreidlin–Wentzell quasipotentialrelative entropy

MSC classification

Primary: 60F10: Large deviations
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82C22: Interacting particle systems 90B15: Network models, stochastic
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 44
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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