arXiv subjects

Thomas Normand

Publications and source records attributed to Thomas Normand.

5 recordsLinked to original sources

arXiv2026

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

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.

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arXiv2026

The narrow escape problem in arbitrary dimension

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 the asymptotic expansion 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.

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arXiv2024

Spectral analysis of a semiclassical random walk associated to a general confining potential

We consider a semiclassical random walk with respect to a probability measure associated to a potential with a finite number of critical points. We recover the spectral results from [1] on the corresponding operator in a more general setting and with improved accuracy. In particular we do not make any assumption on the distribution of the critical points of the potential, in the spirit of [15]. Our approach consists in adapting the ideas from [15] to the recent gaussian quasimodes framework which appears to be more robust than the usual methods, especially when dealing with non local operators.

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arXiv2023

Spectral asymptotics and metastability for the linear relaxation Boltzmann equation

We consider the linear relaxation Boltzmann equation in a semiclassical framework. We construct a family of sharp quasimodes for the associated operator which yields sharp spectral asymptotics for its small spectrum in the low temperature regime. We deduce some information on the long time behavior of the solutions with a sharp estimate on the return to equilibrium as well as a quantitative metastability result. The main novelty is that the collision operator is a pseudo-differential operator in the critical class S^1/2 and that its action on the gaussian quasimodes yields a superposition of exponentials.

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arXiv2022

Metastability results for a class of linear Boltzmann equations

We consider a semiclassical linear Boltzmann model with a non local collision operator. We provide sharp spectral asymptotics for the small spectrum in the low temperature regime from which we deduce the rate of return to equilibrium as well as a metastability result. The main ingredients are resolvent estimates obtained via hypocoercive techniques and the construction of sharp Gaussian quasimodes through an adaptation of the WKB method.

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