Pathwise uniqueness in infinite dimension under weak structure condition

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 dX(t) = AX(t)dt + LB(X(t))dt + GdW(t), t is an element of [0, T], X(0) = x is an element of H (1) where A: D(A) subset of H -> H is a linear operator and the infinitesimal generator of a strongly continuous semigroup {e(tA)}(t >= q), W = {W(t)}(t >= 0) is a V-cylindrical Wiener process defined on a normal filtered probability space (Omega, F, {F-t}(t is an element of[0,T]), P), B : H -> H is a bounded and theta-Holder continuous function, for some suitable theta is an element of (0, 1), and.f : H -> H and G : V -> 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 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.