Approximation of the random inertial manifold of singularly perturbed stochastic wave equations

TL;DR

This paper proves that the random inertial manifold of a singularly perturbed stochastic wave equation can be almost surely approximated by that of a stochastic heat equation, extending the Smolukowski--Kramers approximation to infinite time.

Contribution

It introduces a novel approximation of the inertial manifold for singularly perturbed stochastic wave equations using a stochastic heat equation, applicable almost surely over infinite time.

Findings

Approximation holds almost surely for large time.

The approximation depends on the singular perturbation parameter.

Extension of Smolukowski--Kramers approximation to infinite time.

Abstract

By applying Rohlin's result on the classification of homomorphisms of Lebesgue space, the random inertial manifold of a stochastic damped nonlinear wave equations with singular perturbation is proved to be approximated almost surely by that of a stochastic nonlinear heat equation which is driven by a new Wiener process depending on the singular perturbation parameter. This approximation can be seen as the Smolukowski--Kramers approximation as time goes to infinity. However, as time goes infinity, the approximation changes with the small parameter, which is different from the approximation on a finite interval.