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Small time asymptotics of spectral heat content of isotro...
Rohan Sarkar · 2025-12-09 · via math.PR updates on arXiv.org

The spectral heat content of a domain $Ω\subset\mathbb{R}^d$ corresponding to a $d$-dimensional stochastic process $X=(X_t)_{t\ge 0}$ is defined as \[Q^{X}_Ω(t)=\int_{\mathbb{R}^d} \mathbb{P}_x(τ^X_Ω>t)dx,\] where $τ^X_Ω$ is the first exit time of $X$ from $Ω$. 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 Lévy processes and Gaussian processes, and provide spectral heat content asymptotics for a class of $α$-stable Lévy 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.