Freidlin-Wentzell collision-laws between self-stabilizing diffusions
arXiv:2607.27932
The paper analyzes how two independent self‑stabilizing (McKean‑Vlasov) diffusions in a bistable potential collide when the driving Brownian noise becomes very small, showing that the near‑collision time grows exponentially and collisions occur in specific spatial regions, using Freidlin‑Wentzell exit‑time techniques.
Abstract
The present work investigates the asymptotic behaviours at the zero-noise limit of the first near collision-time and first near collision-location between a pair of independent -dimensional Brownian-driven self-stabilizing (McKean-Vlasov type) diffusions. These asymptotic are considered in a peculiar setting where the systems evolve in a bi-stable landscape and collisions are only triggered by the combined action of the Brownian noises. As the Brownian perturbations fade away, we show that the near collision-time increases at an explicit exponential rate and that related collision-locations persist in specific regions of the space. These results are mainly derived by tailoring classical Freidlin-Wentzell's exit-time (and exit-location) estimates into collision estimates. Similar asymptotic are established for related mean-field interacting particle system approximation, and for the one-dimensional case (where true collisions can be examined
The present paper features a completely revised and augmented version of the unpublished paper: arXiv:2206.04542