On the passage times of self-similar Gaussian processes on curved boundaries
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918178041364480 |
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| author | Khoshnevisan, Davar Lee, Cheuk Yin |
| author_facet | Khoshnevisan, Davar Lee, Cheuk Yin |
| contents | Let $T_{c,β}$ denote the smallest $t\ge1$ that a continuous, self-similar Gaussian process with self-similarity index $α>0$ moves at least $\pm c t^β$ units. We prove that: (i) If $β>α$, then $T_{c,β}=\infty$ with positive probability; (ii) If $β<α$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then $T_{c,β}$ has moments of all order; and (iii) If $β=α$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then there exists a continuous, strictly decreasing function $λ:(0\,,\infty)\to(0\,,\infty)$ such that $\mathrm{E}(T_{c,β}^μ)$ is finite when $0<μ<λ(c)$ and infinite when $μ>λ(c)$. Together these results extend a celebrated theorem of Breiman (1967) and Shepp (1967) for passage times of a Brownian motion on the critical square-root boundary. We briefly discuss two examples: One about fractional Brownian motion, and another about a family of linear stochastic partial differential equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the passage times of self-similar Gaussian processes on curved boundaries Khoshnevisan, Davar Lee, Cheuk Yin Probability Let $T_{c,β}$ denote the smallest $t\ge1$ that a continuous, self-similar Gaussian process with self-similarity index $α>0$ moves at least $\pm c t^β$ units. We prove that: (i) If $β>α$, then $T_{c,β}=\infty$ with positive probability; (ii) If $β<α$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then $T_{c,β}$ has moments of all order; and (iii) If $β=α$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then there exists a continuous, strictly decreasing function $λ:(0\,,\infty)\to(0\,,\infty)$ such that $\mathrm{E}(T_{c,β}^μ)$ is finite when $0<μ<λ(c)$ and infinite when $μ>λ(c)$. Together these results extend a celebrated theorem of Breiman (1967) and Shepp (1967) for passage times of a Brownian motion on the critical square-root boundary. We briefly discuss two examples: One about fractional Brownian motion, and another about a family of linear stochastic partial differential equations. |
| title | On the passage times of self-similar Gaussian processes on curved boundaries |
| topic | Probability |
| url | https://arxiv.org/abs/2506.15949 |