Barriers of the McKean-Vlasov energy via a mountain pass theorem in the space of probability measures

摘要

We show that the empirical process associated with a system of weakly interacting diffusion processes exhibits a form of noise-induced metastability. The result is based on an analysis of the associated McKean-Vlasov free energy, which, for suitable attractive interaction potentials, has at least two distinct global minimisers at the critical parameter value beta = beta(c). On the torus, one of these states is the spatially homogeneous constant state, and the other is a clustered state. We show that a third critical point exists at this value. As a result, we obtain that the probability of transition of the empirical process from the constant state scales like exp(-N Delta), with Delta the energy gap at beta = beta(c). The proof is based on a version of the mountain pass theorem for lower semicontinuous and lambda-geodesically convex functionals on the space of probability measures P-2(M) equipped with the 2-Wasserstein metric, where M is a complete, connected, and smooth Riemannian manifold. (C) 2020 Elsevier Inc. All rights reserved.