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| Auteur principal: | |
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| Format: | Preprint |
| Publié: |
2025
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2512.08595 |
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Table des matières:
- 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.