  A coupled forward-backward stochastic differential system (FBSDS) is
formulated in spaces of fields for the incompressible Navier-Stokes equation in
the whole space. It is shown to have a unique local solution, and further if
either the Reynolds number is small or the dimension of the forward stochastic
differential equation is equal to two, it can be shown to have a unique global
solution. These results are shown with probabilistic arguments to imply the
known existence and uniqueness results for the Navier-Stokes equation, and thus
provide probabilistic formulas to the latter. Related results and the maximum
principle are also addressed for partial differential equations (PDEs) of
Burgers' type. Moreover, from truncating the time interval of the above FBSDS,
approximate solution is derived for the Navier-Stokes equation by a new class
of FBSDSs and their associated PDEs; our probabilistic formula is also bridged
to the probabilistic Lagrangian representations for the velocity field, given
by Constantin and Iyer (Commun. Pure Appl. Math. 61: 330--345, 2008) and Zhang
(Probab. Theory Relat. Fields 148: 305--332, 2010) ; finally, the solution of
the Navier-Stokes equation is shown to be a critical point of controlled
forward-backward stochastic differential equations.
