Small time asymptotics of spectral heat contents for subordinate killed Brownian motions related to isotropic {\alpha}-stable processes

TL;DR

This paper derives detailed small time asymptotic expansions for the spectral heat content of subordinate killed Brownian motions in bounded domains, extending known results to all  and higher-order terms in certain cases.

Contribution

It provides the first comprehensive two-term expansions for all  and three-term expansions for  with ; these results generalize previous work on spectral heat content asymptotics.

Findings

Two-term small time expansion for all  in all dimensions.

Three-term expansion for  when  and dimension .

Extension of spectral heat content asymptotics to subordinate killed Brownian motions.

Abstract

In this paper we study the small time asymptotic behavior of the spectral heat content $\widetilde{Q}_D^{(\alpha)}(t)$ of an arbitrary bounded $C^{1,1}$ domain $D$ with respect to the \textit{subordinate killed Brownian motion} in $D$ via an $(\alpha/2)$-stable subordinator. For all $\alpha\in (0,2)$, we establish a two-term small time expansion for $\widetilde{Q}_D^{(\alpha)}(t)$ in all dimensions. When $\alpha\in (1,2)$ and $d\geq 2$, we establish a three-term small time expansion for $\widetilde{Q}_D^{(\alpha)}(t)$.