Spectral heat content for {\alpha}-stable processes in C1,1 open sets

TL;DR

This paper analyzes the small-time asymptotic behavior of spectral heat content for isotropic alpha-stable processes in smooth bounded domains, extending previous results to all alpha in (0,2) and dimensions d ≥ 1.

Contribution

It establishes the second-term asymptotics of spectral heat content for all alpha in (0,2) and dimensions d ≥ 1, resolving a prior conjecture.

Findings

Derived the second-term asymptotics for spectral heat content in C^{1,1} domains.

Unified results across all alpha in (0,2) and dimensions d ≥ 1.

Resolved the conjecture from previous work on spectral heat content.

Abstract

In this paper we study the asymptotic behavior, as $t\downarrow 0$, of the spectral heat content $Q^{(\alpha)}_{D}(t)$ for isotropic $\alpha$-stable processes, $\alpha\in [1,2)$, in bounded $C^{1,1}$ open sets $D\subset \R^{d}$, $d\geq 2$. Together with the results from \cite{Val2017} for $d=1$ and \cite{GPS19} for $\alpha\in (0,1)$, the main theorem of this paper establishes the asymptotic behavior of the spectral heat content up to the second term for all $\alpha\in (0,2)$ and $d\geq1$, and resolves the conjecture raised in \cite{Val2017}.