Dichotomy in the small-time asymptotics of spectral heat content for L\'evy processes

Abstract

We establish a dichotomy in the small-time asymptotic behavior of the spectral heat content (SHC) for symmetric, but not necessarily isotropic, L\'evy processes whose L\'evy density satisfies a weak lower scaling condition near zero. This dichotomy is governed by whether the process has unbounded or bounded variation. In the unbounded variation case, the leading asymptotic behavior of the SHC is determined by the expected supremum of the process projected in the normal direction near the boundary. In contrast, for processes with bounded variation, the SHC decays linearly in time. Our main result, Theorem \ref{thm:main}, extends and unifies key results from \cite{GPS19}, \cite{KP24}, and \cite{PS22}, covering a broader class of non-isotropic L\'evy processes and offering a streamlined proof.