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Invariant submanifolds for solutions to rough differential equations

Stefan Tappe

TL;DR

This work establishes necessary and sufficient tangency conditions for the local invariance of finite-dimensional submanifolds in Banach spaces under rough differential equations driven by rough paths. It shows that local invariance for truly rough driving signals is equivalent to drift and diffusion tangency to the manifold, with a drift correction term depending on the rough path's second-level data, and that closed manifolds are globally invariant. The results are specialized to random RDEs driven by infinite-dimensional $Q$-Wiener processes and $Q$-fractional Brownian motion, yielding a drift representation $f_0(y) - \tfrac12 \sum_{k=1}^{\infty} \lambda_k Df_k(y) f_k(y)$ and corollaries for Itô/Stratonovich lifts. Overall, the paper provides a pathwise invariant-manifold framework for RDEs and links to SPDE invariance theory, with implications for stochastic and rough dynamical systems.

Abstract

In this paper we provide necessary and sufficient conditions for invariance of finite dimensional submanifolds for rough differential equations (RDEs) with values in a Banach space. Furthermore, we apply our findings to the particular situation of random RDEs driven by $Q$-Wiener processes and random RDEs driven by $Q$-fractional Brownian motion.

Invariant submanifolds for solutions to rough differential equations

TL;DR

This work establishes necessary and sufficient tangency conditions for the local invariance of finite-dimensional submanifolds in Banach spaces under rough differential equations driven by rough paths. It shows that local invariance for truly rough driving signals is equivalent to drift and diffusion tangency to the manifold, with a drift correction term depending on the rough path's second-level data, and that closed manifolds are globally invariant. The results are specialized to random RDEs driven by infinite-dimensional -Wiener processes and -fractional Brownian motion, yielding a drift representation and corollaries for Itô/Stratonovich lifts. Overall, the paper provides a pathwise invariant-manifold framework for RDEs and links to SPDE invariance theory, with implications for stochastic and rough dynamical systems.

Abstract

In this paper we provide necessary and sufficient conditions for invariance of finite dimensional submanifolds for rough differential equations (RDEs) with values in a Banach space. Furthermore, we apply our findings to the particular situation of random RDEs driven by -Wiener processes and random RDEs driven by -fractional Brownian motion.
Paper Structure (6 sections, 31 theorems, 75 equations)

This paper contains 6 sections, 31 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Presentation of the main result
  3. Infinite dimensional random rough differential equations
  4. Rough paths and rough differential equations
  5. Finite dimensional submanifolds in Banach spaces
  6. Functions in Banach spaces

Key Result

Theorem 2.3

We assume that $X$ is truly rough and that $[\mathbf{X}]_t = x t$, $t \in [0,T]$ for some symmetric element $x \in V \otimes V$. Then the following statements are equivalent: If the two equivalent statements (i) and (ii) are fulfilled and the submanifold $\mathcal{M}$ is closed as a subset of $W$, then $\mathcal{M}$ is even globally invariant for the RDE (RDE).

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Corollary 2.6
  • proof : Proof of Theorem \ref{['thm-main']}
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 48 more