Heat content for convolution semigroups

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.