Small Ball Probabilities for the Stochastic Heat Equation on Compact Manifolds
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917228411092992 |
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| author | Chen, Jiaming |
| author_facet | Chen, Jiaming |
| contents | We consider the stochastic heat equation on a compact smooth Riemannian manifold without boundary satisfying \begin{equation*} \partial_tu(t,x)=\frac{1}{2}Δ_Mu(t,x)+σ(t,x,u)\dot{W}(t,x),\quad (t,x)\in\mathbb{R}_+\times M, \end{equation*} where $\dot{W}$ is a centered Gaussian noise that 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 $u(0,x)\equiv 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20794 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Small Ball Probabilities for the Stochastic Heat Equation on Compact Manifolds Chen, Jiaming Probability Primary, 60H15, Secondary, 58J65 We consider the stochastic heat equation on a compact smooth Riemannian manifold without boundary satisfying \begin{equation*} \partial_tu(t,x)=\frac{1}{2}Δ_Mu(t,x)+σ(t,x,u)\dot{W}(t,x),\quad (t,x)\in\mathbb{R}_+\times M, \end{equation*} where $\dot{W}$ is a centered Gaussian noise that 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 $u(0,x)\equiv 0$. |
| title | Small Ball Probabilities for the Stochastic Heat Equation on Compact Manifolds |
| topic | Probability Primary, 60H15, Secondary, 58J65 |
| url | https://arxiv.org/abs/2601.20794 |