Survival probability (heat content) and the lowest eigenvalue of Dirichlet Laplacian

Abstract

We study the survival probability of a particle diffusing in a two-dimensional domain, bounded by a smooth absorbing boundary. The short-time expansion of this quantity depends on the geometric characteristics of the boundary, whilst its long-time asymptotics is governed by the lowest eigenvalue of the Dirichlet Laplacian defined on the domain. We present a simple algorithm for calculation of the short-time expansion for an arbitrary "star-shaped" domain. The coefficients are expressed in terms of powers of boundary curvature, integrated around the circumference of the domain. Based on this expansion, we look for a Pad\'e interpolation between the short-time and the long-time behavior of the survival probability, i.e. between geometric characteristics of the boundary and the lowest eigenvalue of the Dirichlet Laplacian.