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

TL;DR

This paper investigates the small-mass limit of constrained stochastic wave equations with nonlinear conservative noise, deriving a deterministic limit system that preserves the constraint and incorporates noise interactions.

Contribution

It extends the small-mass limit analysis to constrained wave equations with nonlinear multiplicative noise, revealing how noise structure influences the limiting dynamics.

Findings

Derived a deterministic limit system on the unit sphere.

Identified additional terms in the limit due to noise structure.

Showed the limit preserves the energy functional and constraint.

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.