On a Theorem by A.S. Cherny for Semilinear Stochastic Partial Differential Equations

TL;DR

This paper extends Cherny's theorem to infinite-dimensional semilinear stochastic PDEs, establishing equivalences between different notions of uniqueness and linking strong and weak solution concepts.

Contribution

It generalizes Cherny's theorem to Banach space-valued solutions and introduces a new proof using cylindrical martingale problems.

Findings

Weak uniqueness is equivalent to weak joint uniqueness.

A dual version of the Yamada-Watanabe theorem is established.

The proof employs cylindrical martingale problems instead of Cherny's original method.

Abstract

We consider analytically weak solutions to semilinear stochastic partial differential equations with non-anticipating coefficients driven by cylindrical Brownian motion. The solutions are allowed to take values in general separable Banach spaces. We show that weak uniqueness is equivalent to weak joint uniqueness, and thereby generalize a theorem by A.S. Cherny to an infinite dimensional setting. Our proof for the technical key step is different from Cherny's and uses cylindrical martingale problems. As an application, we deduce a dual version of the Yamada-Watanabe theorem, i.e. we show that strong existence and weak uniqueness imply weak existence and strong uniqueness.