PUB - Publikationen an der Universität Bielefeld

We consider the following quasi-linear parabolic system of backward partial differential equations on a Banach space E: (partial derivative t + L)u + f(., ., u, A(1/2)del u) = 0 on [0, T] x E, u(T) = phi, where L is a possibly degenerate second-order differential operator with merely measurable coefficients. We solve this system in the framework of generalized Dirichlet forms and employ the stochastic calculus associated to the Markoy process with generator L to obtain a probabilistic representation of the solution u by solving the corresponding backward stochastic differential equation. The solution satisfies the corresponding mild equation which is equivalent to being a generalized solution to the PDE. A further main result is the generalization of the martingale representation theorem in infinite dimension using the stochastic calculus associated to the generalized Dirichlet form given by L. The tic -linear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to del u.