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

TL;DR

This paper uncovers a fundamental dichotomy in the small-time asymptotic behavior of spectral heat content for symmetric Le9vy processes, distinguishing cases based on whether the process has bounded or unbounded variation.

Contribution

It extends and unifies previous results by establishing a broader, more general framework for the small-time asymptotics of spectral heat content in non-isotropic Le9vy processes.

Findings

Unbounded variation processes' SHC asymptotics depend on the expected supremum near the boundary.

Bounded variation processes' SHC decays linearly with time.

The paper generalizes previous results and provides a streamlined proof.

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.