Measure-dependent non-linear diffusions with superlinear drifts: asymptotic behaviour of the first exit-times

Abstract

In this paper, we study McKean-Vlasov SDE living in ${\mathbb{R}}^d$ in the reversible case without assuming any type of convexity assumptions for confinement or interaction potentials. Kramers' type law for the exit-time from a domain of attraction is established. Namely, in the small-noise regime, the limit in probability of the first exit-time behaves exponentially. This result is established using the large deviations principle as well as improved coupling method. Having removed the convexity assumption, this work is a major improvement of the previously known results for the exit-time problem, the review of which is provided in the paper.

36 pages, 4 figures