A spectral approach to the narrow escape problem in two-dimensional domains

Louis Carillo; Tony Lelièvre; Thomas Normand; Urbain Vaes

A spectral approach to the narrow escape problem in two-dimensional domains

Analysis of PDEs 2026-08-05 v1 Probability

Authors: Louis Carillo , Tony Lelièvre , Thomas Normand , Urbain Vaes

Abstract

We study the law of the exit time and exit point of a Brownian motion in a two-dimensional domain with reflecting boundary conditions, except on small disjoint exit windows through which the stochastic process can escape the domain. In the limit of infinitely small exit windows, it is natural to assume that the process starts from the quasi-stationary distribution. In this setting, we obtain a precise description of the exit event.

Keywords

narrow escape problem Brownian motion quasi-stationary distribution exit time exit point spectral method

Cite

@article{arxiv.2608.05007,
  title  = {A spectral approach to the narrow escape problem in two-dimensional domains},
  author = {Louis Carillo and Tony Lelièvre and Thomas Normand and Urbain Vaes},
  journal= {arXiv preprint arXiv:2608.05007},
  year   = {2026}
}