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

Rehmeier M. On Cherny's results in infinite dimensions: a theorem dual to Yamada-Watanabe. Stochastics and Partial Differential Equations: Analysis and Computations . 2021;9:33-70.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 + s(t, X)dWt, t = 0, 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, 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