A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE

We extend some results on time-homogeneous processes generated by divergence form operators to time-inhomogeneous ones. These results concern the decomposition of such processes as Dirichlet process, with an explicit expression for the term of zero-quadratic variation. Moreover, we extend some results on the Itô formula and BSDEs related to weak solutions of PDEs, and we study the case of quasi-linear PDEs. Finally, our results are used to prove the existence of weak solutions to forward-backward stochastic differential equations.