Weak solutions to stochastic differential equations with discontinuous drift in Hilbert space. Stochastic processes, physics and geometry: new interplays, II

Professor Sergio Albeverio has been interested in solutions of infinite dimensional stochastic differential equations with values in C([0, T], R ^{z d} ) motivated by applications to physics ([1], [2]). Leha and Ritter in [6] have studied the existence problem in C([0, T], H ^w ). Our purpose here is to extend the latter work by weakening the assumptions on the potential, using techniques developed in our earlier work. More precisely, we study the problem of the existence of weak solutions for SDE's with discontinuous drift in a Hilbert space H. The discontinuity is modeled by a countable family of real valued functions. The solution has finite dimensional Galerkin approximation and is realized in the space of continuous functions with values in H, where H is endowed with its weak topology. Th.is work extends a result in [6] and also shows that under the assumptions in [6] both, the Galerkin approximation and the infinite dimensional approximation of [6] produce solutions with identical laws.