On the well-posedness of SPDEs with locally Lipschitz coefficients
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909784802852864 |
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| author | Foondun, Mohammud Khoshnevisan, Davar Nualart, Eulalia |
| author_facet | Foondun, Mohammud Khoshnevisan, Davar Nualart, Eulalia |
| contents | We consider the stochastic partial differential equation, $\partial_t u = \tfrac12 \partial^2_x u + b(u) + σ(u) \dot{W},$ where $u=u(t\,,x)$ is defined for $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, and $\dot{W}$ denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition $u(0)$ is bounded and measurable, and $b$ and $σ$ are locally Lipschitz continuous functions and have at most linear growth. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The results naturally generalize to the case where $b$ and $σ$ are time dependent with uniform-in-time growth and oscillation properties. Additionally, our method can be extended to the stochastic wave equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09381 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the well-posedness of SPDEs with locally Lipschitz coefficients Foondun, Mohammud Khoshnevisan, Davar Nualart, Eulalia Probability We consider the stochastic partial differential equation, $\partial_t u = \tfrac12 \partial^2_x u + b(u) + σ(u) \dot{W},$ where $u=u(t\,,x)$ is defined for $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, and $\dot{W}$ denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition $u(0)$ is bounded and measurable, and $b$ and $σ$ are locally Lipschitz continuous functions and have at most linear growth. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The results naturally generalize to the case where $b$ and $σ$ are time dependent with uniform-in-time growth and oscillation properties. Additionally, our method can be extended to the stochastic wave equation. |
| title | On the well-posedness of SPDEs with locally Lipschitz coefficients |
| topic | Probability |
| url | https://arxiv.org/abs/2411.09381 |