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

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 \(L^2\)-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-Itô correction term.