We consider analytically weak solutions to semilinear stochastic partial differential equations with non-anticipating coefficients driven by a cylindrical Brownian motion. The solutions are allowed to take values in Banach spaces. We show that weak uniqueness is equivalent to weak joint uniqueness, and thereby generalize a theorem by A.S. Cherny to an infinite dimensional setting. Our proof for the technical key step is different from Cherny’s and uses cylindrical martingale problems. As an application, we deduce a dual version of the Yamada–Watanabe theorem, i.e. we show that strong existence and weak uniqueness imply weak existence and strong uniqueness.

我们考虑具有非预期系数的半线性随机偏微分方程的解析弱解，该方程由圆柱布朗运动驱动。允许解在 Banach 空间中取值。我们证明弱唯一性等同于弱联合唯一性，从而将 AS Cherny 的定理推广到无限维设置。我们对技术关键步骤的证明与 Cherny 的不同，并使用圆柱鞅问题。作为应用，我们推导出Yamada-Watanabe 定理的对偶版本，即我们证明强存在性和弱唯一性意味着弱存在性和强唯一性。