The smallmass limit for some constrained wave equations with nonlinear conservative noise
Abstract
We study the smallmass limit, also known as the SmoluchowskiKramers 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 squareintegrable 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2024
 DOI:
 10.48550/arXiv.2409.08021
 arXiv:
 arXiv:2409.08021
 Bibcode:
 2024arXiv240908021C
 Keywords:

 Mathematics  Probability