Pathwise uniqueness by noise for singular stochastic PDEs

TL;DR

This paper develops a new theoretical framework for establishing pathwise uniqueness in singular stochastic PDEs with drift, addressing a key open problem and covering applications from fluid dynamics to phase separation.

Contribution

It introduces a self-contained theory for stochastic evolution equations on Hilbert spaces, providing the first results on pathwise uniqueness for singular stochastic PDEs with drift.

Findings

Achieves novel uniqueness results for fluid-dynamics models

Addresses phase-separation models with singular drift

Provides a foundational framework for future research

Abstract

Pathwise uniqueness for stochastic PDEs with drift in differential form is a main open problem in the recent literature on regularisation by noise. This paper establishes a self-contained theory in the framework of stochastic evolution equations on separable Hilbert spaces and provides a first result to address such an issue. The singularity of the drift allows to achieve novel uniqueness results for several classes of examples, ranging from fluid-dynamics to phase-separation models.