Heat kernels of non-symmetric jump processes with exponentially decaying jumping kernel

TL;DR

This paper investigates the transition densities of non-symmetric jump processes with exponentially decaying kernels, providing sharp bounds under certain conditions, advancing understanding of their probabilistic behavior.

Contribution

It introduces new sharp bounds for transition densities of non-symmetric jump processes with exponential or subexponential decay, under specific scaling conditions.

Findings

Upper bounds decay at the same rate as jumping kernels

Lower bounds are sharp under weak upper scaling conditions

Provides comprehensive analysis of transition densities for these processes

Abstract

In this paper we study the transition densities for a large class of non-symmetric Markov processes whose jumping kernels decay exponentially or subexponentially. We obtain their upper bounds which also decay at the same rate as their jumping kernels. When the lower bounds of jumping kernels satisfy the weak upper scaling condition at zero, we also establish lower bounds for the transition densities, which are sharp.