Stochastic Wave Equations with Constraints: Well-Posedness and Smoluchowski–Kramers Diffusion Approximation

• Published in: Communications in mathematical physics Vol. 406; no. 9
• Main Authors: Brzeźniak, Zdzisław, Cerrai, Sandra
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.09.2025
• Subjects: Approximation; Boundary conditions; Constraints; Euclidean geometry; Hilbert space; Wave equations; Well posed problems
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-025-05397-0

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 lies 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 $$L^2$$ L 2 -norm of the solution is equal to one. We introduce a small mass $$\mu >0$$ 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.