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

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