Smoluchowski-Kramers approximation for the damped stochastic wave equation with multiplicative noise in any spatial dimension

TL;DR

This paper proves that solutions to the damped stochastic wave equation with multiplicative noise converge to the stochastic heat equation in any spatial dimension, extending previous results to more general noise types and dimensions.

Contribution

It establishes the validity of the Smoluchowski-Kramers approximation for multiplicative noise in all spatial dimensions, broadening the scope of prior work.

Findings

Convergence of solutions in any spatial dimension.

Extension of approximation to multiplicative noise.

Validation of the Smoluchowski-Kramers approximation in new settings.

Abstract

We show that the solutions to the damped stochastic wave equation converge pathwise to the solution of a stochastic heat equation. This is called the Smoluchowski-Kramers approximation. Cerrai and Freidlin have previously demonstrated that this result holds in the cases where the system is exposed to additive noise in any spatial dimension or when the system is exposed to multiplicative noise and the spatial dimension is one. The current paper proves that the Smoluchowski-Kramers approximation is valid in any spatial dimension when the system is exposed to multiplicative noise.