Path-dependent PDE and Backward SDE

In this talk we present a new type of Soblev norm defined in the space of functions of continuous paths. Under the Wiener probability measure the corresponding norm is suitable to prove the existence and uniqueness for a large type of system of path dependent quasi-linear parabolic partial differential equations (PPDE). We have establish 1-1 correspondence between this new type of PPDE and the classical backward SDE (BSDE). For fully nonlinear PPDEs, the corresponding Sobolev norm is under a sublinear expectation called G-expectation, in the place of Wiener expectation. The canonical process becomes a new type of nonlinear Brownian motion called G-Brownian motion. A similar 1-1 correspondence has been established. We can then apply the recent results of existence, uniqueness and principle of comparison for BSDE driven by G-Brownian motion to obtain the same result for the PPDE.