Points of slow growth for parabolic SPDEs
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866912770766667776 |
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| author | Khoshnevisan, Davar Lee, Cheuk Yin |
| author_facet | Khoshnevisan, Davar Lee, Cheuk Yin |
| contents | Consider the stochastic PDE, $\partial_tu = \partial^2_x u + σ(u) \dot{W}$ on $\mathbb{R}_+\times\mathbb{R}$, subject to $u(0)\equiv1$, where $\dot{W}$ denotes space-time white noise on $\mathbb{R}_+\times\mathbb{R}$ and $σ:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous. It is known that $u(t\,,x)-1$ has approximately a Gaussian distribution for every $x$ when $t\approx0$. Here we prove that there exist random points $x\in\mathbb{R}$ where the fluctuations of the solution near times zero are almost surely of sharp order $t^{1/4}$. Our work bears some loose resemblance to the study of the slow points of Brownian motion increments, though significant challenges arise due to the infinite-dimensional nature of the present problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15177 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Points of slow growth for parabolic SPDEs Khoshnevisan, Davar Lee, Cheuk Yin Probability Consider the stochastic PDE, $\partial_tu = \partial^2_x u + σ(u) \dot{W}$ on $\mathbb{R}_+\times\mathbb{R}$, subject to $u(0)\equiv1$, where $\dot{W}$ denotes space-time white noise on $\mathbb{R}_+\times\mathbb{R}$ and $σ:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous. It is known that $u(t\,,x)-1$ has approximately a Gaussian distribution for every $x$ when $t\approx0$. Here we prove that there exist random points $x\in\mathbb{R}$ where the fluctuations of the solution near times zero are almost surely of sharp order $t^{1/4}$. Our work bears some loose resemblance to the study of the slow points of Brownian motion increments, though significant challenges arise due to the infinite-dimensional nature of the present problem. |
| title | Points of slow growth for parabolic SPDEs |
| topic | Probability |
| url | https://arxiv.org/abs/2512.15177 |