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

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