BSDE and generalized Dirichlet forms: The infinite dimensional case

We consider the following quasi-linear parabolic system of backward partial differential equations on a Banach space E: (∂_t + L)u + f(·, ·, u; A^1/2∇u) = 0 on [0, T] × E, u_T = φ, 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 Markov 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 non-linear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to ∇u.