Stochastic invariance in infinite dimension beyond Lipschitz coefficients

TL;DR

This paper provides new necessary and sufficient conditions for stochastic invariance of closed sets in infinite-dimensional Hilbert spaces for SDE solutions, extending classical results to low-regularity coefficients.

Contribution

It introduces two novel characterizations of invariance involving normal vectors and the positive maximum principle, applicable under mild regularity assumptions.

Findings

Characterization using normal vectors and Moore-Penrose pseudoinverse

Extension of Stratonovich correction to low-regularity coefficients

Application to invariant manifolds in Hilbert spaces

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.