Quasi-stationarity and metastability: Insights from weakly interacting particle systems and applications to SPDE

Abstract. Many models from physics and the life sciences exhibit metastable behaviour -- that is, behaviour that appears to be stable over some time scale, but dies out or significantly changes on longer time scales. For instance, consider a system of N particles moving as Brownian motions interacting via an attractive potential (such as the Glauber dynamics associated with the classical mean-field O(2) spin system of statistical mechanics). 
In the large particle limit of such systems, the empirical measure of such systems is known to converge to a nonlocal parabolic PDE of McKean-Vlasov type.  While the McKean-Vlasov system is known to possess no non-trivial stationary solutions, numerical experiments demonstrate the existence of an almost-synchronized state that persists over a long time scale. 
In this talk, we characterize this almost-synchronized state and the time scale on which it persists using methods involving sub-Markov semigroups, quasi-stationary distributions, and the spectral theory of Schrödinger operators applied to the finite particle system. Control on the time scale in terms of the noise amplitude and particle number are obtained in total variation and Wasserstein distance. Time permitting, we will discuss how the insights gained from the study of metastability in many particle systems may lead to the application of similar methods to metastable behaviour in SPDE.