Small Ball Probabilities for the Fractional Stochastic Heat Equation Driven by a Colored Noise
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915165005414400 |
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| author | Chen, Jiaming |
| author_facet | Chen, Jiaming |
| contents | We consider the fractional stochastic heat equation on the $d$-dimensional torus $\mathbb{T}^d:=\left[-\frac{1}{2},\frac{1}{2}\right]^d$, $d\geq 1$, with periodic boundary conditions: \[ \partial_t u(t,\textbf{x})= -(-Δ)^{α/2}u(t,\textbf{x})+σ(t,\textbf{x},u)\dot{F}(t,\textbf{x})\quad \textbf{x}\in \mathbb{T}^d,t\in\mathbb{R}_+ ,\] where $α\in(1,2]$ and $\dot{F}(t,\textbf{x})$ is a generalized Gaussian noise which is white in time and colored in space. Assuming that $σ$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,\textbf{x})\equiv 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_13781 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Small Ball Probabilities for the Fractional Stochastic Heat Equation Driven by a Colored Noise Chen, Jiaming Probability We consider the fractional stochastic heat equation on the $d$-dimensional torus $\mathbb{T}^d:=\left[-\frac{1}{2},\frac{1}{2}\right]^d$, $d\geq 1$, with periodic boundary conditions: \[ \partial_t u(t,\textbf{x})= -(-Δ)^{α/2}u(t,\textbf{x})+σ(t,\textbf{x},u)\dot{F}(t,\textbf{x})\quad \textbf{x}\in \mathbb{T}^d,t\in\mathbb{R}_+ ,\] where $α\in(1,2]$ and $\dot{F}(t,\textbf{x})$ is a generalized Gaussian noise which is white in time and colored in space. Assuming that $σ$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,\textbf{x})\equiv 0$. |
| title | Small Ball Probabilities for the Fractional Stochastic Heat Equation Driven by a Colored Noise |
| topic | Probability |
| url | https://arxiv.org/abs/2207.13781 |