Small time asymptotics of spectral heat contents for subordinate killed Brownian motions related to isotropic α-stable processes

Abstract

In this paper we study the small time asymptotic behavior of the spectral heat content ${\tilde{Q}}_D^{(\alpha )}(t)$ 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 ${\tilde{Q}}_D^{(\alpha )}(t)$ in all dimensions. When $\alpha \in (1,2)$ and $d\ge 2$, we establish a three-term small time expansion for ${\tilde{Q}}_D^{(\alpha )}(t)$.