# Stochastic wave equations with constraints: well-posedness and Smoluchowski-Kramers diffusion approximation

## Abstract:

I will discuss 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 on the unit sphere.

A specific example is provided by the stochastic damped wave equation in a bounded domain of a $d$-dimensional Euclidean space, endowed with

the Dirichlet boundary conditions, 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 the 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.

This talk is based on joint research with S. Cerrai (Maryland).