Heat content for convolution semigroups

Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a Lévy process in $\mathbb{R}^d$ and $Ω$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H (t) = \int_Ω\mathbb{P}_{x} (X_t\in Ω^c) dx$ which is called the heat content. We study its asymptotic behaviour as $t$ goes to zero for isotropic Lévy processes under some mild assumptions on the characteristic exponent. We also treat the class of Lévy processes with finite variation in full generality.

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