Stochastic wave equations with constraints: well-posedness and Smoluchowski-Kramers diffusion approximation
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909665794719744 |
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| author | Cerrai, Sandra Brzeźniak, Zdzislaw |
| author_facet | Cerrai, Sandra Brzeźniak, Zdzislaw |
| contents | We investigate the well-posedness of a class of stochastic second-order in time damped evolution equations in Hilbert spaces, subject to the constraint that the solution lie within the unitary sphere. Then, we focus on a specific example, the stochastic damped wave equation in a bounded domain of a $d$-dimensional Euclidean space, endowed with the Dirichlet boundary condition, with the added constraint that the $L^2$-norm of the solution is equal to one. We introduce a small mass $μ>0$ in front of the second-order derivative in time and examine the validity of a Smoluchowski-Kramers diffusion approximation. We demonstrate that, in the small mass limit, the solution converges to the solution of a stochastic parabolic equation subject to the same constraint. We further show that an extra noise-induced drift emerges, which in fact does not account for the Stratonovich-to-Itô correction term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_09717 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stochastic wave equations with constraints: well-posedness and Smoluchowski-Kramers diffusion approximation Cerrai, Sandra Brzeźniak, Zdzislaw Probability We investigate the well-posedness of a class of stochastic second-order in time damped evolution equations in Hilbert spaces, subject to the constraint that the solution lie within the unitary sphere. Then, we focus on a specific example, the stochastic damped wave equation in a bounded domain of a $d$-dimensional Euclidean space, endowed with the Dirichlet boundary condition, with the added constraint that the $L^2$-norm of the solution is equal to one. We introduce a small mass $μ>0$ in front of the second-order derivative in time and examine the validity of a Smoluchowski-Kramers diffusion approximation. We demonstrate that, in the small mass limit, the solution converges to the solution of a stochastic parabolic equation subject to the same constraint. We further show that an extra noise-induced drift emerges, which in fact does not account for the Stratonovich-to-Itô correction term. |
| title | Stochastic wave equations with constraints: well-posedness and Smoluchowski-Kramers diffusion approximation |
| topic | Probability |
| url | https://arxiv.org/abs/2303.09717 |