Reducing exit-times of diffusions with repulsive interactions

¹ Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, 44000 Nantes, France
² School of Mathematics, University of Birmingham, B15 2TT Birmingham, UK
³ LJLL and LCT, Sorbonne Université, 4 place Jussieu, 75005 Paris, France
⁴ CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
⁵ Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France

^* Corresponding author: h.duong@bham.ac.uk

Received: 2 September 2022
Accepted: 14 June 2023

Abstract

In this work we prove a Kramers’ type law for the low-temperature behavior of the exittimes from a metastable state for a class of self-interacting nonlinear diffusion processes. Contrary to previous works, the interaction is not assumed to be convex, which means that this result covers cases where the exit-time for the interacting process is smaller than the exit-time for the associated non-interacting process. The technique of the proof is based on the fact that, under an appropriate contraction condition, the interacting process is conveniently coupled with a non-interacting (linear) Markov process where the interacting law is replaced by a constant Dirac mass at the fixed point of the deterministic zero-temperature process.