## 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

[

BibTeX]

*preprint*

**Inserted:** 3 aug 2019

**Year:** 2019

**Abstract:**

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 $\mathbb{R}^d$.