Weak uniqueness for stochastic partial differential equations in Hilbert spaces

Let $U,H$ be two separable Hilbert spaces. The main goal of this paper is to study the weak uniqueness of the Stochastic Differential Equation evolving in $H$ \begin{align*} dX(t)=AX(t)dt+$\mathcal{V}$B(X(t))dt+GdW(t), \quad t>0, \quad X(0)=x \in H, \end{align*} where $\{W(t){\}}_{t\ge 0}$ is a $U$-cylindrical Wiener process, $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup, $\mathcal{V},G:U\to H$ are linear bounded operators and $B:H\to U$ is a uniformly continuous function. The abstract result in this paper gives the weak uniqueness for large classes of heat and damped equations in any dimension without any Hölder continuity assumption on $B$.