Small time asymptotics of spectral heat content of isotropic processes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909995123081216 |
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| author | Sarkar, Rohan |
| author_facet | Sarkar, Rohan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08595 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Small time asymptotics of spectral heat content of isotropic processes Sarkar, Rohan Probability 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. |
| title | Small time asymptotics of spectral heat content of isotropic processes |
| topic | Probability |
| url | https://arxiv.org/abs/2512.08595 |