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 d X t = b ( t , X ) d t + σ ( t , X ) d W t , 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.