A note on invariant manifolds for stochastic partial differential equations in the framework of the variational approach

TL;DR

This paper establishes conditions under which finite-dimensional submanifolds remain invariant for solutions of SPDEs within the variational framework, linking these to SPDEs in embedded spaces.

Contribution

It introduces new conditions for local invariance of finite-dimensional submanifolds in SPDEs using the variational approach, connecting to SPDEs in embedded spaces.

Findings

Provides criteria for local invariance of submanifolds in SPDEs

Connects invariance conditions to SPDEs in embedded spaces

Enhances understanding of geometric properties of SPDE solutions

Abstract

In this note we provide conditions for local invariance of finite dimensional submanifolds for solutions to stochastic partial differential equations (SPDEs) in the framework of the variational approach. For this purpose, we provide a connection to SPDEs in continuously embedded spaces.