Small time asymptotics of spectral heat content of isotropic processes

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.