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.
Keywords
Cite
@article{arxiv.1105.2626,
title = {Survival probability (heat content) and the lowest eigenvalue of Dirichlet Laplacian},
author = {P. Kalinay and L. Samaj and I. Travenec},
journal= {arXiv preprint arXiv:1105.2626},
year = {2011}
}
Comments
Accepted in IJMPB