The Steklov problem for exterior domains: asymptotic behavior and applications

TL;DR

This paper studies the spectral properties of the Steklov problem for the modified Helmholtz equation in exterior domains, analyzing asymptotic behaviors and applications to stochastic processes and probability densities.

Contribution

It provides the first detailed asymptotic analysis of Steklov eigenvalues and eigenfunctions for the modified Helmholtz equation in exterior domains, with applications to stochastic processes.

Findings

Asymptotic behavior of eigenvalues and eigenfunctions for small parameter p

Application to long-time behavior of probability densities in stochastic processes

Validation of theoretical results using finite-element numerical methods

Abstract

We investigate the spectral properties of the Steklov problem for the modified Helmholtz equation $(p-\Delta) u = 0$ in the exterior of a compact set, for which the positive parameter $p$ ensures exponential decay of the Steklov eigenfunctions at infinity. We obtain the small-$p$ asymptotic behavior of the eigenvalues and eigenfunctions and discuss their features for different space dimensions. These results find immediate applications to the theory of stochastic processes and unveil the long-time asymptotic behavior of probability densities of various first-passage times in exterior domains. Theoretical results are validated by solving the exterior Steklov problem by a finite-element method with a transparent boundary condition.