Heat content for convolution semigroups

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. 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 Lévy processes under some mild assumptions on the characteristic exponent. We also treat the class of Lévy processes with finite variation in full generality.

The article has been accepted for publication in J. Math. Anal. Appl

References in corpus (2)

• Asymptotic behavior of densities of unimodal convolution semigroups
• Heat content for stable processes in domains of ${\mathbb{R}}^d$

Cited by in corpus (4)

• On the domain of fractional Laplacians and related generators of Feller processes
• Spectral Heat Content for Lévy Processes
• On the heat content for the poisson kernels over the unit ball in the euclidean space
• Heat content estimates for the fractional Schrödinger operator $\F+\ind$