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

Abstract

AbstractWe 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 $$\begin{aligned} \text {d}X_t=b(t,X)\text {d}t+\sigma (t,X)\text {d}W_t, \quad t\ge 0, \end{aligned}$$ 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 $$V \subseteq H \subseteq E$$ 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.