Time-dependent averages of a critical long-range stochastic heat equation
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908134305431552 |
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| author | Kuzgun, Sefika Tao, Ran |
| author_facet | Kuzgun, Sefika Tao, Ran |
| contents | We study the time-dependent spatial averages of a critical stochastic partial differential equation, namely the stochastic heat equation in dimension $d\geq 3$ with noise white in time and colored in space with covariance kernel $\|\cdot\|^{-2}$. The solution to this SPDE is a singular measure and was constructed by Mueller and Tribe in [MT04]. We show that the time-dependent spatial averages of this SPDE over a ball of radius $R$ at time $t$ have different limits under different space-time scales. In particular, when $t\ll R^2$, the central limit theorem holds; when $t=R^2$, the spatial average is a non-Gaussian random variable; when $t\gg R^2$, the spatial average becomes extinct. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09058 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Time-dependent averages of a critical long-range stochastic heat equation Kuzgun, Sefika Tao, Ran Probability We study the time-dependent spatial averages of a critical stochastic partial differential equation, namely the stochastic heat equation in dimension $d\geq 3$ with noise white in time and colored in space with covariance kernel $\|\cdot\|^{-2}$. The solution to this SPDE is a singular measure and was constructed by Mueller and Tribe in [MT04]. We show that the time-dependent spatial averages of this SPDE over a ball of radius $R$ at time $t$ have different limits under different space-time scales. In particular, when $t\ll R^2$, the central limit theorem holds; when $t=R^2$, the spatial average is a non-Gaussian random variable; when $t\gg R^2$, the spatial average becomes extinct. |
| title | Time-dependent averages of a critical long-range stochastic heat equation |
| topic | Probability |
| url | https://arxiv.org/abs/2411.09058 |