Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift

TL;DR

This paper establishes strong pathwise uniqueness for stochastic evolution equations in Hilbert spaces with bounded measurable drift and cylindrical Wiener noise, extending finite-dimensional results to infinite dimensions using Malliavin calculus techniques.

Contribution

It generalizes Veretennikov's finite-dimensional uniqueness result to infinite-dimensional Hilbert spaces with minimal regularity assumptions.

Findings

Proves strong uniqueness for a broad class of stochastic evolution equations in Hilbert spaces.

Employs Malliavin-Sobolev space methods suitable for infinite-dimensional analysis.

Shows limitations on initial distributions for which uniqueness holds.

Abstract

We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov's fundamental result on $\mathbb{R}^d$ to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin-Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. Such restriction, however, is common in infinite dimensions.