Effective dynamics of interfaces for nonlinear SPDEs driven by multiplicative white noise

TL;DR

This paper derives effective interface dynamics for nonlinear SPDEs with multiplicative white noise, simplifying complex infinite-dimensional problems into finite-dimensional SDEs, and verifies their equivalence under small noise conditions.

Contribution

It introduces a detailed method to approximate the slow manifold of nonlinear SPDEs with multiplicative noise using finite-dimensional SDEs, providing a new tool for analyzing stochastic interface dynamics.

Findings

Derived effective equations on slow manifolds for nonlinear SPDEs

Verified equivalence between full SPDEs and reduced SDE systems under small noise

Applied results to four stochastic models demonstrating practical relevance

Abstract

In the present work, we investigate the dynamics of the infinite-dimensional stochastic partial differential equation (SPDE) with multiplicative white noise. We derive the effective equation on the approximate slow manifold in detail by utilizing a finite-dimensional stochastic differential equation (SDE) describing the motion of interfaces. In particular, we verify the equivalence between the full SPDE and the coupled system under small stochastic perturbations. Moreover, we apply our results to effective dynamics of stochastic models with multiplicative white noise, illustrated with four examples on the stochastic damped wave equation, the stochastic Allen-Cahn equation, the stochastic nonlinear Schr\"odinger equation and the stochastic Swift-Hohenberg equation.