Asymptotics for the expected lifetime of Brownian motion on thin domains in R^n

TL;DR

This paper develops detailed asymptotic expansions for the expected lifetime of Brownian motion on thin domains in R^n, providing insights into how domain geometry influences stochastic processes and related physical quantities.

Contribution

It introduces new three-term and two-term asymptotic expansions for expected lifetime and torsional rigidity on thin domains, extending previous eigenvalue analysis methods.

Findings

Derived three-term asymptotic expansion for expected lifetime

Obtained two-term expansion for maximum expected lifetime

Showed expansions can approximate exit times for less scaled domains

Abstract

We derive a three-term asymptotic expansion for the expected lifetime of Brownian motion and for the torsional rigidity on thin domains in R^n, and a two-term expansion for the maximum (and corresponding maximizer) of the expected lifetime. The approach is similar to that which we used previously to study the eigenvalues of the Dirichlet Laplacian and consists of scaling the domain in one direction and deriving the corresponding asymptotic expansions as the scaling parameter goes to zero. Apart from being dominated by the one-dimensional Brownian motion along the direction of the scaling, we also see that the symmetry of the perturbation plays a role in the expansion. As in the case of eigenvalues, these expansions may also be used to approximate the exit time for domains where the scaling parameter is not necessarilly close to zero.