Estimates of heat kernels of non-symmetric Lévy processes

We investigate densities of vaguely continuous convolution semigroups of probability measures on ℝ d {{\mathbb{R}^{d}}} . First, we provide results that give upper estimates in a situation when the corresponding jump measure is allowed to be highly non-symmetric. Further, we prove upper estimates of the density and its derivatives if the jump measure compares with an isotropic unimodal measure and the characteristic exponent satisfies a certain scaling condition. Lower estimates are discussed in view of a recent development in that direction, and in such a way to complement upper estimates. We apply all those results to establish precise estimates of densities of non-symmetric Lévy processes.

Asymptotic Behaviour and Estimates of Slowly Varying Convolution Semigroups

We study the asymptotic formulas and estimates for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R}^{d}$ under the assumption that its Lévy–Khintchine exponent varies slowly. We derive some new estimates of the transition densities and Green functions.