Intrinsic compound kernel estimates for the transition probability density of a L\'evy type processes and their applications

TL;DR

This paper develops estimates for the transition probability density of Lévy-type processes using intrinsic compound kernels, providing tools for analyzing measures within Kato and Dynkin classes.

Contribution

It introduces a novel method to construct fundamental solutions and derive intrinsic estimates for Lévy-type processes, advancing the understanding of their transition densities.

Findings

Constructed fundamental solutions for specific integro-differential equations.

Derived intrinsic upper and lower bounds for these solutions.

Provided criteria for measure classification within Kato and Dynkin classes.

Abstract

In this paper we construct the fundamental solution to some integro-differential equation, as well as the intrinsic upper and lower estimates for this solution. As an application of constructed estimates we state a criterion when a given Borel measure belongs to the respective Kato and Dynkin classes.