L{é}vy processes: concentration function and heat kernel bounds

Abstract

We investigate densities of vaguely continuous convolution semigroups of probability measures on ${\mathbb{R}}^d$. We expose that many typical conditions on the characteristic exponent repeatedly used in the literature of the subject are equivalent to the behaviour of the maximum of the density as a function of time variable. We also prove qualitative lower estimates under mild assumptions on the corresponding jump measure and the characteristic exponent.

arXiv admin note: substantial text overlap with arXiv:1710.07793