A support theorem for parabolic stochastic PDEs with nondegenerate Hölder diffusion coefficients
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915260005351424 |
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| author | Han, Yi |
| author_facet | Han, Yi |
| contents | In this paper we work with parabolic SPDEs of the form
$$ \partial_t u(t,x)=\partial_x^2 u(t,x)+g(t,x,u)+σ(t,x,u)\dot{W}(t,x)
$$ with Neumann boundary conditions, where $x\in[0,1]$, $\dot{W}(t,x)$ is the space-time white noise on $(t,x)\in[0,\infty)\times [0,1]$, $g$ is uniformly bounded, and the solution $u\in\mathbb{R}$ is real valued. The diffusion coefficient $σ$ is assumed to be uniformly elliptic but only Hölder continuous in $u$. Previously, support theorems for SPDEs have only been established assuming that $σ$ is Lipschitz continuous in $u$. We obtain new support theorems and small ball probabilities in this $σ$ Hölder continuous case via the recently established sharp two sided estimates of stochastic integrals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_00502 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A support theorem for parabolic stochastic PDEs with nondegenerate Hölder diffusion coefficients Han, Yi Probability In this paper we work with parabolic SPDEs of the form $$ \partial_t u(t,x)=\partial_x^2 u(t,x)+g(t,x,u)+σ(t,x,u)\dot{W}(t,x) $$ with Neumann boundary conditions, where $x\in[0,1]$, $\dot{W}(t,x)$ is the space-time white noise on $(t,x)\in[0,\infty)\times [0,1]$, $g$ is uniformly bounded, and the solution $u\in\mathbb{R}$ is real valued. The diffusion coefficient $σ$ is assumed to be uniformly elliptic but only Hölder continuous in $u$. Previously, support theorems for SPDEs have only been established assuming that $σ$ is Lipschitz continuous in $u$. We obtain new support theorems and small ball probabilities in this $σ$ Hölder continuous case via the recently established sharp two sided estimates of stochastic integrals. |
| title | A support theorem for parabolic stochastic PDEs with nondegenerate Hölder diffusion coefficients |
| topic | Probability |
| url | https://arxiv.org/abs/2302.00502 |