Bsde and generalized dirichlet forms: The finite-dimensional case

We consider the following quasi-linear parabolic system of backward partial differential equations (∂ +L )u+f(.,.,u,∇_uσ)= 0 on [0,T] × R^d 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 of the PDE. A further main result is the generalization of the martingale representation theorem using the stochastic calculus associated to the generalized Dirichlet form given by L. The nonlinear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to ∇.