Let V, H be two separable Hilbert spaces, and \(T>0\) . We consider a stochastic differential equation which evolves in the Hilbert space H of the form 1 \(\begin{aligned} dX(t)=AX(t)dt+{\mathscr {L}}B(X(t))dt+GdW(t), \quad t\in [0,T], \quad X(0)=x \in H, \end{aligned}\) where \(A:D(A)\subseteq H\rightarrow H\) is a linear operator and the infinitesimal generator of a strongly continuous semigroup \(\{e^{tA}\}_{t\ge 0}\) , \(W=\{W(t)\}_{t\ge 0}\) is a V-cylindrical Wiener process defined on a normal filtered probability space \((\Omega ,\mathcal {F},\{\mathcal {F}_t\}_{t\in [0,T]},\mathbb {P})\) , \(B:H\rightarrow H\) is a bounded and \(\theta \) -Hölder continuous function, for some suitable \(\theta \in (0,1)\) , and \({\mathscr {L}}:H\rightarrow H\) and \(G:V\rightarrow H\) are linear bounded operators. We prove that, under suitable assumptions on the coefficients, the weak mild solution to equation (1) depends on the initial datum in a Lipschitz way. This implies that for (1), pathwise uniqueness holds. Here, the presence of the operator \({\mathscr {L}}\) plays a crucial role. In particular, the conditions assumed on the coefficients cover the stochastic damped wave equation in dimension 1 and the stochastic damped Euler–Bernoulli Beam equation up to dimension 3 even in the hyperbolic case.
