A note on the generalized heat content for Lévy processes

Abstract

Let $\mathbf{X}=\{X_t{\}}_{t\ge 0}$ be a Lévy process in ${\mathbb{R}}^d$ and $\Omega$ be an open subset of ${\mathbb{R}}^d$ with finite Lebesgue measure. The quantity $H(t)={\int }_{\Omega }{\mathbb{P}}^x(X_t\in {\Omega }^c)dx$ is called the heat content. In this article we consider its generalized version $H_g^{\mu }(t)={\int }_{{\mathbb{R}}^d}{\mathbb{E}}^{x}g(X_t)\mu (dx)$, where $g$ is a bounded function and $\mu$ a finite Borel measure. We study its asymptotic behaviour at zero for various classes of Lévy processes.

arXiv commentto appear in the Bulletin of Korean Mathematical Society