Small time asymptotics of spectral heat content of isotropic processes

TL;DR

This paper develops a new method to analyze the small time behavior of spectral heat content for isotropic processes, extending existing results and applying to time-changed stable Lévy processes with connections to non-local Cauchy problems.

Contribution

A novel technique for small time asymptotics of spectral heat content applicable to translation invariant isotropic processes, including stable Lévy and Gaussian processes.

Findings

Recovered known results for Lévy and Gaussian processes

Derived spectral heat content asymptotics for time-changed stable Lévy processes

Connected spectral heat content asymptotics to non-local Cauchy problems

Abstract

The spectral heat content of a domain $\Omega\subset\mathbb{R}^d$ corresponding to a $d$-dimensional stochastic process $X=(X_t)_{t\ge 0}$ is defined as \[Q^{X}_\Omega(t)=\int_{\mathbb{R}^d} \mathbb{P}_x(\tau^X_\Omega>t)dx,\] where $\tau^X_\Omega$ is the first exit time of $X$ from $\Omega$. We provide a novel technique for proving small time asymptotic of spectral heat content for any translation invariant isotropic process satisfying negligible tail probability condition. As a consequence, we recover several existing results in the context of L\'evy processes and Gaussian processes, and provide spectral heat content asymptotics for a class of $\alpha$-stable L\'evy processes time-changed by right inverse of positive, increasing, self-similar Markov processes. The latter has connection to some Cauchy problems that are non-local in both time and space.