Parametrix construction for certain L\'evy-type processes

TL;DR

This paper develops a parametrix method for certain Lévy-type processes, establishing the generator's properties, transition density bounds, and smoothness, with illustrative examples.

Contribution

It introduces a parametrix construction for non-local operators associated with Lévy-type processes, extending their analysis and providing explicit bounds and smoothness results.

Findings

The non-local operator extends to a Markov process generator.

Explicit upper and lower bounds for the transition density are derived.

Smoothness properties of the transition density are established.

Abstract

In this paper we show that a non-local operator of certain type extends to the generator of a strong Markov process, admitting the transition probability density. For this transition probability density we construct the intrinsic upper and lower bounds, and prove some smoothness properties. Some examples are provided.