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Stochastic wave equations with constraints: well-posedness and Smoluchowski-Kramers diffusion approximation by Cerrai, Sandra; Brzeźniak, Zdzislaw is a scholarly article available to read on EtoBox.

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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 $\mu>0$ 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-It\^{o} correction term.

Author

Cerrai, Sandra; Brzeźniak, Zdzislaw

Published

2023

Language

EN