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 (Formula presented.) where A:D(A)⊆H→H is a linear operator and the infinitesimal generator of a strongly continuous semigroup {etA}t≥0, W={W(t)}t≥0 is a V-cylindrical Wiener process defined on a normal filtered probability space (Ω,F,{Ft}t∈[0,T],P), B:H→H is a bounded and θ-Hölder continuous function, for some suitable θ∈(0,1), and L: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.