Calculus of Variations and Geometric Measure Theory

Rishabh S. Gvalani - A. Schlichting

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

created by gvalani on 03 Aug 2019



Inserted: 3 aug 2019

Year: 2019

ArXiv: 1905.11823 PDF


We show that the empirical process associated to 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 $\mathcal{P}(M)$ equipped with the $W_2$ Wasserstein metric, where $M$ is a Riemannian manifold or $\mathbb{R}^d$.