The narrow escape problem in arbitrary dimension
Abstract
The narrow escape problem is a prototypical example for studying entropic metastability, motivated by the analysis of biological and chemical systems. The problem concerns the determination of the exit time and position of a Brownian particle trapped in a domain with a reflecting boundary pierced by narrow holes. Our goal is to investigate this problem in a general domain in any dimension (greater than or equal to two), using the quasi-stationary distribution approach to metastability. In particular, we derive first order asymptotic expansions with error bounds of the mean exit time and the law of the exit position in the limit where the hole sizes tend to zero. Our analytical predictions are illustrated by numerical simulations, using dedicated Monte Carlo techniques.
Related
- A spectral approach to the narrow escape problem in two-dimensional domains
- The Two-Dimensional Narrow Escape Problem with Deterministic Initial Condition: Mean Escape Time and Escape location probabilities
- The recent counterexamples to the Eisenbud-Schreyer-Weyman Ulrich problem (2003) and to Hochster's small Cohen-Macaulay conjecture (1973): a survey, with a generalization to arbitrary dimension
- Narrow escape problems beyond spherical holes: slits and tubes
- Irreducibility of the Bloch variety for periodic Schrödinger operators in arbitrary dimension
BibTeX
@misc{carillo2026narrow,
title = {{The narrow escape problem in arbitrary dimension}},
author = {Louis Carillo and Tony Lelièvre and Thomas Normand and Gabriel Stoltz and Urbain Vaes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.02212},
eprint = {2608.02212},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02212},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The narrow escape problem in arbitrary dimension
AU - Louis Carillo
AU - Tony Lelièvre
AU - Thomas Normand
AU - Gabriel Stoltz
AU - Urbain Vaes
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2608.02212
UR - https://arxiv.org/abs/2608.02212
AB - The narrow escape problem is a prototypical example for studying entropic metastability, motivated by the analysis of biological and chemical systems. The problem concerns the determination of the exit time and position of a Brownian particle trapped in a domain with a reflecting boundary pierced by narrow holes. Our goal is to investigate this problem in a general domain in any dimension (greater than or equal to two), using the quasi-stationary distribution approach to metastability. In particular, we derive first order asymptotic expansions with error bounds of the mean exit time and the law of the exit position in the limit where the hole sizes tend to zero. Our analytical predictions are illustrated by numerical simulations, using dedicated Monte Carlo techniques.
ER -