Invariant manifolds for random and stochastic partial differential equations

TL;DR

This paper establishes the existence of random invariant manifolds for a class of stochastic partial differential equations under nonuniform hyperbolicity conditions, extending the geometric understanding of complex stochastic dynamics.

Contribution

It introduces a novel approach to construct random pseudo-stable and pseudo-unstable manifolds without using random norms, applicable to nonlinear SPDEs with linear multiplicative noise.

Findings

Existence of random pseudo-stable and pseudo-unstable manifolds under nonuniform hyperbolicity.

Application to nonlinear SPDEs with linear multiplicative noise.

Extension of invariant manifold theory beyond stochastic ODEs.

Abstract

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable and pseudo-unstable manifolds for a class of \emph{random} partial differential equations and \emph{stochastic} partial differential equations is shown. Unlike the invariant manifold theory for stochastic \emph{ordinary} differential equations, random norms are not used. The result is then applied to a nonlinear stochastic partial differential equation with linear multiplicative noise.