The small-mass limit for some constrained wave equations with nonlinear conservative noise

Abstract

We study the small-mass limit, also known as the Smoluchowski-Kramers diffusion approximation (see \cite{kra} and \cite{smolu}), for a system of stochastic damped wave equations, whose solution is constrained to live in the unitary sphere of the space of square-integrable functions on the interval $(0,L)$. The stochastic perturbation is given by a nonlinear multiplicative Gaussian noise, where the stochastic differential is understood in Stratonovich sense. Due to its particular structure, such noise not only conserves $\mathbb{P}$-a.s. the constraint, but also preserves a suitable energy functional. In the limit, we derive a deterministic system, that remains confined to the unit sphere of $L^2$, but includes additional terms. These terms depend on the reproducing kernel of the noise and account for the interaction between the constraint and the particular conservative noise we choose.