Asymptotic expansion of the nonlocal heat content

TL;DR

This paper investigates the asymptotic behavior of the heat content associated with Lévy processes, including isotropic stable and more general types, providing insights into their boundary-related properties as time approaches zero.

Contribution

It offers a new asymptotic expansion for the heat content of Lévy processes, extending previous results to more general processes under mild conditions.

Findings

Derived asymptotic expansion for isotropic α-stable Lévy processes.

Extended analysis to general Lévy processes with mild assumptions.

Provided insights into boundary behavior of Lévy processes.

Abstract

Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a L\'evy process in $\mathbb{R}^d$ and $\Omega$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H(t)=\int_{\Omega} \mathbb{P}^x (X_t\in\Omega^c) \, \mathrm{d}x$ which is called the heat content. We study its asymptotic expansion for isotropic $\alpha$-stable L\'evy processes and more general L\'evy processes, under mild assumptions on the characteristic exponent.