{"title":"The small-mass limit for some constrained wave equations with nonlinear conservative noise","authors":"Sandra Cerrai, Mengzi Xie","doi":"arxiv-2409.08021","DOIUrl":null,"url":null,"abstract":"We study the small-mass limit, also known as the Smoluchowski-Kramers\ndiffusion approximation (see \\cite{kra} and \\cite{smolu}), for a system of\nstochastic damped wave equations, whose solution is constrained to live in the\nunitary sphere of the space of square-integrable functions on the interval\n$(0,L)$. The stochastic perturbation is given by a nonlinear multiplicative\nGaussian noise, where the stochastic differential is understood in Stratonovich\nsense. Due to its particular structure, such noise not only conserves\n$\\mathbb{P}$-a.s. the constraint, but also preserves a suitable energy\nfunctional. In the limit, we derive a deterministic system, that remains\nconfined to the unit sphere of $L^2$, but includes additional terms. These\nterms depend on the reproducing kernel of the noise and account for the\ninteraction between the constraint and the particular conservative noise we\nchoose.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"12 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08021","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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.