A note on the generalized heat content for L\'evy processes

TL;DR

This paper investigates the asymptotic behavior at zero of a generalized heat content associated with Lévy processes, extending classical results to broader classes of processes and functions.

Contribution

It introduces a generalized heat content for Lévy processes and analyzes its asymptotics at zero, broadening understanding beyond classical heat content.

Findings

Asymptotic behavior characterized for various Lévy processes

Extension of classical heat content results to generalized versions

Insights into the influence of process and measure choices on heat content

Abstract

Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a L\'{e}vy process in $\mathbb{R}^d$ and $\Omega$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. The quantity $H (t) = \int_{\Omega} \mathbb{P}^{x} (X_t\in \Omega ^c) d x$ is called the heat content. In this article we consider its generalized version $H_g^\mu (t) = \int_{\mathbb{R}^d}\mathbb{E}^{x} g(X_t)\mu( d x )$, where $g$ is a bounded function and $\mu$ a finite Borel measure. We study its asymptotic behaviour at zero for various classes of L\'{e}vy processes.