Invariant manifolds for stochastic partial differential equations in continuously embedded Hilbert spaces

TL;DR

This paper characterizes when finite-dimensional submanifolds remain invariant under solutions of SPDEs in Hilbert spaces and links this to invariance in finite-dimensional SDEs, offering new analytical tools.

Contribution

It provides necessary and sufficient conditions for stochastic invariance of submanifolds in SPDEs and establishes a novel connection to finite-dimensional SDE invariance analysis.

Findings

Necessary and sufficient conditions for invariance in SPDEs.

Link between invariance in Hermite Sobolev spaces and finite-dimensional SDEs.

New invariance results for finite-dimensional SDEs.

Abstract

We provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional It\^{o} diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs.