Estimates of heat kernels of non-symmetric L\'evy processes

TL;DR

This paper derives upper and lower bounds for the densities of non-symmetric Lévy processes, advancing understanding of their heat kernels and providing precise estimates under various conditions.

Contribution

It introduces new upper and lower density estimates for non-symmetric Lévy processes, including derivatives, under specific scaling and symmetry conditions.

Findings

Upper estimates for densities with highly non-symmetric jump measures

Bounds on derivatives of densities when measures compare with isotropic unimodal measures

Complementary lower estimates for density bounds

Abstract

We investigate densities of vaguely continuous convolution semigroups of probability measures on $\mathbb{R}^d$. 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 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 L\'evy processes.