A note on the generalized heat content for Lévy processes

Abstract

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. The quantity $H (t) = \int_Ω \mathbb{P}^{x} (X_t\in Ω^c) d x$ is called the heat content. In this article we consider its generalized version $H_g^μ(t) = \int_{\mathbb{R}^d}\mathbb{E}^{x} g(X_t)μ( d x )$, where $g$ is a bounded function and $μ$ a finite Borel measure. We study its asymptotic behaviour at zero for various classes of Lévy processes.