On Cherny’s results in infinite dimensions: a theorem dual to Yamada–Watanabe

We prove that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions to stochastic partial differential equations of type dXt=b(t,X)dt+σ(t,X)dWt,t≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {d}X_t=b(t,X)\text {d}t+\sigma (t,X)\text {d}W_t, \quad t\ge 0, \end{aligned}$$\end{document}and show that for such equations uniqueness in law is equivalent to joint uniqueness in law for deterministic initial conditions. Here W is a cylindrical Wiener process in a separable Hilbert space U and the equation is considered in a Gelfand triple V⊆H⊆E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V \subseteq H \subseteq E$$\end{document}, where H is some separable (infinite-dimensional) Hilbert space. This generalizes the corresponding results of Cherny, who proved these statements for the case of finite-dimensional equations.