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

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 \(V \subseteq H \subseteq E\), 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.

我们证明，在法律上联合唯一性和强解的存在意味着类型为随机偏微分方程变分解的路径唯一性

并表明对于此类方程，对于确定性初始条件，法律的唯一性等同于法律的联合唯一性。这里W是可分离的希尔伯特空间U中的圆柱维纳过程，方程在Gelfand三元组（（V \ subseteq H \ subseteq E \）中考虑，其中H是一些可分离的（无限维）希尔伯特空间。这概括了Cherny的相应结果，Cherny在有限​​维方程的情况下证明了这些陈述。