Heat content for convolution semigroups

Abstract

Let X={Xt}t≥0 be a Lévy process in Rd and Ω be an open subset of Rd with finite Lebesgue measure. In this article we consider the quantity H(t)=∫ΩPx(Xt∈Ωc)dx related to X 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.