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We prove that joint uniqueness in law and the existence of a strong solutionimply pathwise uniqueness for variational solutions to stochastic partialdifferential equations of the form \begin{align*}\text{d}X_t=b(t,X)\text{d}t+\sigma(t,X)\text{d}W_t, \,\,\,t\geq 0, \end{align*}and show that for such equations uniqueness in law is equivalent to jointuniqueness in law. Here $W$ is a cylindrical Wiener process in a separableHilbert space $U$ and the equation is considered in a Gelfand triple $V\subseteq H \subseteq E$, where $H$ is some separable (infinite-dimensional)Hilbert space. This generalizes the corresponding results of A. Cherny for thecase of finite-dimensional equations.

On Cherny's results in infinite dimensions: A theorem dual to Yamada-Watanabe