  Forward-backward stochastic differential equations (FBSDEs) have attracted
significant attention since they were introduced almost 30 years ago, due to
their wide range of applications, from solving non-linear PDEs to pricing
American-type options. Here, we consider two new classes of multidimensional
FBSDEs with distributional coefficients (elements of a Sobolev space with
negative order). We introduce a suitable notion of a solution, show existence
and uniqueness of a strong solution of the first FBSDE, and weak existence for
the second. We establish a link with PDE theory via a nonlinear Feynman-Kac
representation formula. The associated semi-linear second order parabolic PDE
is the same for both FBSDEs, also involves distributional coefficients and has
not previously been investigated; our analysis uses mild solutions, Sobolev
spaces and semigroup theory.
