A Kramers' type law for the first collision-time of two self-stabilizing diffusions and of their particle approximations

TL;DR

This paper studies the asymptotic behavior of the first collision-time and location for self-stabilizing diffusions and their particle approximations in a double-well landscape, revealing exponential growth rates as noise vanishes.

Contribution

It establishes a Kramers' type law for collision times in self-stabilizing diffusions, linking collision phenomena to exit-time problems via Freidlin-Wentzell theory, in both 1D and multidimensional cases.

Findings

Collision times grow exponentially as noise vanishes.

Collision locations tend to a specific point in space.

Results apply to both one-dimensional and multidimensional diffusions.

Abstract

The present work investigates the asymptotic behaviors, at the zero-noise limit, of the first collision-time and first collision-location related to a pair of self-stabilizing diffusions and of their related particle approximations. These asymptotic are considered in a peculiar framework where diffusions evolve in a double-wells landscape and where collisions manifest due to the combined action of the Brownian motions driving each diffusion and the action of a self-stabilizing kernel. As the Brownian effects vanish, we show that first collision-times grow at an explicit exponential rate and that the related collision-locations persist at a special point in space. These results are mainly obtained by linking collision phenomena for diffusion processes with exit-time problems of random perturbed dynamical systems, and by exploiting Freidlin-Wentzell's LDP approach to solve these exit-time problems. Importantly, we consider two distinctive situations: the one-dimensional case (where true collisions can be directly studied) and the general multidimensional case (where collisions are required to be enlarged).