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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.08021 |
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| _version_ | 1866909313391394816 |
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| author | Cerrai, Sandra Xie, Mengzi |
| author_facet | Cerrai, Sandra Xie, Mengzi |
| contents | We study the small-mass limit, also known as the Smoluchowski-Kramers diffusion approximation (see \cite{kra} and \cite{smolu}), for a system of stochastic damped wave equations, whose solution is constrained to live in the unitary sphere of the space of square-integrable functions on the interval $(0,L)$. The stochastic perturbation is given by a nonlinear multiplicative Gaussian noise, where the stochastic differential is understood in Stratonovich sense. Due to its particular structure, such noise not only conserves $\mathbb{P}$-a.s. the constraint, but also preserves a suitable energy functional. In the limit, we derive a deterministic system, that remains confined to the unit sphere of $L^2$, but includes additional terms. These terms depend on the reproducing kernel of the noise and account for the interaction between the constraint and the particular conservative noise we choose. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08021 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The small-mass limit for some constrained wave equations with nonlinear conservative noise Cerrai, Sandra Xie, Mengzi Probability We study the small-mass limit, also known as the Smoluchowski-Kramers diffusion approximation (see \cite{kra} and \cite{smolu}), for a system of stochastic damped wave equations, whose solution is constrained to live in the unitary sphere of the space of square-integrable functions on the interval $(0,L)$. The stochastic perturbation is given by a nonlinear multiplicative Gaussian noise, where the stochastic differential is understood in Stratonovich sense. Due to its particular structure, such noise not only conserves $\mathbb{P}$-a.s. the constraint, but also preserves a suitable energy functional. In the limit, we derive a deterministic system, that remains confined to the unit sphere of $L^2$, but includes additional terms. These terms depend on the reproducing kernel of the noise and account for the interaction between the constraint and the particular conservative noise we choose. |
| title | The small-mass limit for some constrained wave equations with nonlinear conservative noise |
| topic | Probability |
| url | https://arxiv.org/abs/2409.08021 |