## UC Berkeley / Lawrence Berkeley Laboratory

#### Smoluchowski-Kramers diffusion-approximation for constrained stochastic wave equations

**Sandra Cerrai, University of Maryland**

We consider a 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 does not account for the Stratonovich-to-It\^{o} correction term.