Abstract. Let X = {X t } t≥0 be a Lévy process in R d and Ω be an open subset of R d with finite Lebesgue measure. In this article we consider the quantity H(t) = Ω P x (X t ∈ 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.
Abstract. We introduce and study a class of random walks defined on the integer lattice Z d -a discrete space and time counterpart of the symmetric α-stable process in R d . When 0 < α < 2 any coordinate axis in Z d , d ≥ 3, is a non-massive set whereas any cone is massive. We provide a necessary and sufficient condition for the thorn to be a massive set.
Abstract. We prove the asymptotic formulas for the densities of isotropic unimodal convolution semigroups of probability measures on R d under the assumption that its Lévy-Khintchine exponent is regularly varying of index between 0 and 2.
Abstract. We consider a random walk Sτ which is obtained from the simple random walk S by a discrete time version of Bochner's subordination. We prove that under certain conditions on the subordinator τ appropriately scaled random walk Sτ converges in the Skorohod space to the symmetric α-stable process B α . We also prove asymptotic formula for the transition function of Sτ similar to the Pólya's asymptotic formula for B α .
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