Relatore:- Zdzislaw Brzezniak (University of York)
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).