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

Abstract

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 $\mathcal{P}_2(M)$ equipped with the $2$-Wasserstein metric, where $M$ is a complete, connected, and smooth Riemannian manifold.