Heat content for convolution semigroups

TL;DR

This paper investigates the small-time asymptotic behavior of heat content for isotropic and finite variation Lévy processes in Euclidean space, providing insights into their probabilistic and analytical properties.

Contribution

It offers a detailed analysis of heat content asymptotics for a broad class of Lévy processes, including isotropic and finite variation cases, under mild assumptions.

Findings

Asymptotic formulas for heat content as time approaches zero.

Extension of results to Lévy processes with finite variation.

Insights into the probabilistic structure 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 $H (t) = \int_{\Omega}\mathbb{P}_{x} (X_t\in \Omega ^c) dx$ which is called the heat content. We study its asymptotic behaviour as $t$ goes to zero for isotropic L\'evy processes under some mild assumptions on the characteristic exponent. We also treat the class of L\'evy processes with finite variation in full generality.