Large Time Behaviour and Metastability in Mean-Field Interacting Particle Systems

Abstract

This thesis studies the large time behaviour and metastability in weakly interacting Markov processes with jumps. Our motivation is to quantify the large time behaviour of various networked systems that arise in practice. The first set of results are for finite-state mean-field interacting particle systems. We first obtain a sharp estimate (in the exponential scale) on the time required for convergence of the empirical measure process of the $N$-particle system to its invariant measure; we show that when time is of the order of $\exp\{N\Lambda\}$ for a suitable constant $\Lambda > 0$, the process has mixed well and it is close to its invariant measure. We then obtain large-$N$ asymptotics of the second largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as $\exp\{-N\Lambda\}$. The main tools used in establishing these results are the large deviation properties of the empirical measure process from its large-$N$ limit. As an application of the study of the large time behaviour, we also show the convergence of the empirical measure of the system of particles to a global minimum of a certain `entropy' function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.

We then consider an extension of this finite-state mean-field model in which the particles are subject to a fast varying random environment. The second result of this thesis is the path-space large deviation principle (LDP) for the joint law of the empirical measure process of the particles and the occupation measure process of the fast environment. This extends previous results known for two time scale diffusions to two time scale mean-field models with jumps. Our proof is based on the method of stochastic exponentials. We characterise the rate function by studying a certain variational problem associated with an exponential martingale.

The third result is on the asymptotics of the invariant measure in countable-state mean-field models. The Freidlin-Wentzell quasipotential is the usual candidate rate function for the sequence of invariant measures indexed by the number of particles. We first provide 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, we study 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.