Trace Estimates for Stable Processes

TL;DR

This paper investigates the small-time behavior of the trace of heat semigroups for symmetric stable processes in R-smooth domains, revealing that the second asymptotic term relates to the domain's surface area, similar to Brownian motion.

Contribution

It extends trace asymptotic analysis from Brownian motion to symmetric stable processes in R-smooth domains, highlighting the surface area term in the second asymptotic component.

Findings

Second term in asymptotics involves surface area

Results apply to R-smooth domains

Generalizes Brownian motion trace estimates

Abstract

In this paper we study the behaviour in time of the trace (the partition function) of the heat semigroup associated with symmetric stable processes in domains of $\Rd$. In particular, we show that for domains with the so called {\it{$R$-smoothness}} property the second terms in the asymptotic as $t\to 0$ involves the surface area of the domain, just as in the case of Brownian motion.