On the trace of unimodal L\'evy processes on Lipschitz domains

TL;DR

This paper establishes that for unimodal Lévy processes on Lipschitz domains, the second term in the heat kernel trace asymptotics as time approaches zero is proportional to the boundary surface area, aligning with symmetric stable process results.

Contribution

It extends the asymptotic analysis of the heat kernel trace to unimodal Lévy processes on Lipschitz domains, showing boundary surface area determines the second term.

Findings

Second term in trace asymptotics equals boundary surface area.

Results align unimodal Lévy processes with symmetric stable processes.

Asymptotic behavior holds under weak scaling conditions.

Abstract

We show that the second term in the asymptotic expansion as t approaches 0 of the trace of the Dirichlet heat kernel on Lipschitz domains for unimodal L\'evy processes, satisfying some weak scaling conditions, is given by the surface area of the boundary of the domain. This brings the asymptotics for the trace of unimodal L\'evy processes in domains of Euclidean space on par with those of symmetric stable processes as far as boundary smoothness is concerned.