A unified approach to the small-time behavior of the spectral heat content for isotropic Lévy processes

Abstract

This paper establishes the precise small-time asymptotic behavior of the spectral heat content for isotropic Lévy processes on bounded $C^{1,1}$ open sets of ${\mathbb{R}}^d$ with $d\ge 2$, where the underlying characteristic exponents are regularly varying at infinity with index $\alpha \in (1,2]$, including the case $\alpha =2$. Moreover, this asymptotic behavior is shown to be stable under an integrable perturbation of its Lévy measure. These results cover a wide class of isotropic Lévy processes, including Brownian motions, stable processes, and jump diffusions, and the proofs provide a unified approach to the asymptotic behavior of the spectral heat content for all of these processes.

To appear in Stochastic Processes and their Applications