The Navier-Stokes-$\alpha$ equation via forward-backward stochastic differential systems

In this paper, we use forward-backward stochastic differential systems to study the solution of two and d dimensional ($d\ge 3$) Navier-Stokes-$\alpha$ equation. For the two dimensional Navier-Stokes-$\alpha$ equation with space periodic boundary conditions, we derive the Feynmann-Kac formula associated with the vorticity equation and prove the global existence and uniqueness of the solution. For the d dimensional ($d\ge 3$) case, we prove the local existence and uniqueness of the solution in Sobolev space.