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

TL;DR

This paper investigates the survival probability of a diffusing particle in a two-dimensional domain, linking short-time boundary geometry to long-time eigenvalues, and introduces an algorithm for calculating these properties in star-shaped domains.

Contribution

It presents a new algorithm for computing short-time expansion coefficients based on boundary curvature for star-shaped domains and explores interpolation between short- and long-time behaviors.

Findings

Derived explicit formulas for short-time expansion coefficients.

Developed a Padé interpolation method connecting boundary geometry and eigenvalues.

Provided a practical computational approach for complex domains.

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.