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).