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

Rajeev Bhaskaran, Stefan Tappe

TL;DR

The paper analyzes local invariance of finite-dimensional submanifolds for SPDEs within the variational framework, connecting variational SPDEs to SPDEs in continuously embedded spaces. It establishes equivalence between invariance criteria in the two frameworks under various assumptions, notably when $K$ is a separable Hilbert space and in the general setting with measurable drift. The results yield concrete applications to Hermite Sobolev spaces and to linear submanifolds generated by eigenfunctions of the $p$-Laplacian, thereby providing practical criteria to identify invariant manifolds and ensure local solvability of SPDEs in these contexts.

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.

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

TL;DR

The paper analyzes local invariance of finite-dimensional submanifolds for SPDEs within the variational framework, connecting variational SPDEs to SPDEs in continuously embedded spaces. It establishes equivalence between invariance criteria in the two frameworks under various assumptions, notably when is a separable Hilbert space and in the general setting with measurable drift. The results yield concrete applications to Hermite Sobolev spaces and to linear submanifolds generated by eigenfunctions of the -Laplacian, thereby providing practical criteria to identify invariant manifolds and ensure local solvability of SPDEs in these contexts.

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.
Paper Structure (8 sections, 15 theorems, 41 equations)

This paper contains 8 sections, 15 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Stochastic partial differential equations in continuously embedded spaces
  3. Stochastic partial differential equations in the framework of the variational approach
  4. The situation where $K$ is a separable Hilbert space
  5. The general situation
  6. Invariant submanifolds in Hermite Sobolev spaces
  7. Linear submanifolds for the stochastic $p$-Laplace equation
  8. Continuously embedded spaces

Key Result

Proposition 2.6

BT-preprint Let $\mathcal{M}$ be an finite dimensional $C^k$-submanifold of $H$. Then the following statements are equivalent:

Theorems & Definitions (39)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • ...and 29 more