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Stochastic invariance in infinite dimension beyond Lipschitz coefficients

Eduardo Abi Jaber, Stefan Tappe

Abstract

We establish necessary and sufficient conditions for stochastic invariance of closed subsets in Hilbert spaces for solutions to infinite-dimensional stochastic differential equations (SDEs) under mild assumptions on the coefficients. Our first characterization is formulated in terms of certain normal vectors to the invariance set and requires differentiability only of the dispersion operator, but not of the diffusion coefficient itself. The condition involves a suitable corrected drift expressed through the dispersion operator and its Moore-Penrose pseudoinverse, extending the classical Stratonovich correction term to the present low-regularity setting. Our second characterization is given in terms of the positive maximum principle for the infinitesimal generator of the associated diffusion process. We illustrate our characterizations in the case of invariant manifolds.

Stochastic invariance in infinite dimension beyond Lipschitz coefficients

Abstract

We establish necessary and sufficient conditions for stochastic invariance of closed subsets in Hilbert spaces for solutions to infinite-dimensional stochastic differential equations (SDEs) under mild assumptions on the coefficients. Our first characterization is formulated in terms of certain normal vectors to the invariance set and requires differentiability only of the dispersion operator, but not of the diffusion coefficient itself. The condition involves a suitable corrected drift expressed through the dispersion operator and its Moore-Penrose pseudoinverse, extending the classical Stratonovich correction term to the present low-regularity setting. Our second characterization is given in terms of the positive maximum principle for the infinitesimal generator of the associated diffusion process. We illustrate our characterizations in the case of invariant manifolds.
Paper Structure (26 sections, 149 theorems, 466 equations)

This paper contains 26 sections, 149 theorems, 466 equations.

Table of Contents

  1. Introduction
  2. Presentation of the main results
  3. The general framework
  4. The infinitesimal generator
  5. The invariance property
  6. The Heine-Borel property
  7. Main results
  8. The convergence of the series
  9. Equivalent characterizations
  10. Invariant manifolds
  11. The necessity proof
  12. Necessity under smoothness on $\sigma$: Theorem \ref{['thm-main-sigma']}.
  13. Necessity under smoothness on $C$: Theorem \ref{['thm-main']}.
  14. The drift condition: The case of distinct eigenvalues
  15. The drift condition: The general case
  16. ...and 11 more sections

Key Result

Lemma 2.1

The following statements are true:

Theorems & Definitions (331)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 321 more