On the existence of weak variational solutions to stochastic differential equations

We study the existence of weak variational solutions in a Gelfand triplet of real separable Hilbert spaces, under continuity, growth, and coer- civity conditions on the coecients of the stochastic dierential equation. The laws of finite dimensional approximations are proved to weakly converge to the limit which is identified as a weak solution. The solution is an H- valued continuous process in L2(›,C((0,T),H)) \ L2((0,T) ◊ ›,V ). Under the assumption of monotonicity the solution is strong and unique. equations in the dual of a nuclear space, and constructed generalized solutions to SPDE's. Our purpose here is to adapt the techniques in (3) to produce a function space weak solution to the variational problem, as in (13), posed in a Gelfand triplet V ,! H ,! V ⁄ , where V and H are real separable Hilbert spaces. The space V ⁄ is the continuous linear dual Hilbert space, and all injections are continuous and dense. The norms and scalar products are denoted by h·,·iV , k · kV , and similar for the spaces H and V ⁄ . The duality on V ◊ V ⁄ is denoted by h·,·i and it agrees with the scalar product in H, i.e. hv,hi = hv,hiH if h 2 H. We use the ideas in (2) and instead of the embeddings being Hilbert-Schmidt operators as in (3), we only assume their compactness. It should be noted that the weak solution X that we construct is in C((0,T),H) with X 2 L2((0,T) ◊ ›,V ), and it satisfies