# Spectral Heat Content for L\'evy Processes

## TL;DR

This paper investigates the asymptotic behavior of spectral heat content for various Lévy processes, revealing stability under perturbations and extending results to different process variations and set types.

## Contribution

It provides new asymptotic results for spectral heat content of Lévy processes of bounded and unbounded variation, including stability under Lévy measure perturbations.

## Key findings

- Asymptotic behavior characterized for bounded variation Lévy processes.
- Spectral heat content analyzed for open sets of finite measure in one dimension.
- Stability of asymptotics under Lévy measure perturbations established.

## Abstract

In this paper we study the spectral heat content for various L\'evy processes. We establish the asymptotic behavior of the spectral heat content for L\'{e}vy processes of bounded variation in $\mathbb{R}^{d}$, $d\geq 1$. We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in $\mathbb{R}$ with respect to L\'{e}vy processes of unbounded variation under certain conditions on their characteristic exponents. Finally we establish that the asymptotic behavior of the spectral heat content is stable under integrable perturbations to the L\'{e}vy measure.