On the Traces of symmetric stable processes on Lipschitz domains

TL;DR

This paper establishes that the second term in the small-time asymptotic expansion of the trace of symmetric stable processes in Lipschitz domains is proportional to the boundary surface area, aligning their behavior with Brownian motion.

Contribution

It extends the asymptotic trace results known for Brownian motion to symmetric stable processes in Lipschitz domains, showing the boundary term depends on surface area.

Findings

Second term in trace asymptotics equals boundary surface area.

Results apply to all symmetric stable processes with 0<α<2.

Aligns stable process asymptotics with Brownian motion in Lipschitz domains.

Abstract

It is shown that the second term in the asymptotic expansion as $t\to 0$ of the trace of the semigroup of symmetric stable processes (fractional powers of the Laplacian) of order $\alpha$, for any $0<\alpha<2$, in Lipschitz domains is given by the surface area of the boundary of the domain. This brings the asymptotics for the trace of stable processes in domains of Euclidean space on par with those of Brownian motion (the Laplacian), as far as boundary smoothness is concerned.