Strong uniqueness for SDEs in Hilbert spaces with nonregular drift

TL;DR

This paper establishes pathwise uniqueness and strong solutions for a class of infinite-dimensional SDEs with nonregular drift, extending classical finite-dimensional results to Hilbert spaces using advanced analytical techniques.

Contribution

It generalizes finite-dimensional uniqueness results to infinite dimensions for SDEs with nonregular drift, employing maximal regularity, Dirichlet forms, and Zvonkin transformations.

Findings

Proves pathwise uniqueness for SDEs in Hilbert spaces with nonregular drift.

Establishes weak existence and strong solutions via Yamada-Watanabe theorem.

Applies results to stochastic reaction diffusion equations with bounded measurable drift.

Abstract

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and nondegenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also prove weak existence, hence obtain unique strong solutions by the Yamada-Watanabe theorem. The proofs are based in part on a recent maximal regularity result in infinite dimensions, the theory of quasi-regular Dirichlet forms and an infinite dimensional version of a Zvonkin-type transformation. As a main application, we show pathwise uniqueness for stochastic reaction diffusion equations perturbed by a Borel measurable bounded drift. Hence, such SDE have a unique strong solution.