Navier-Stokes equations and forward-backward SDEs on the group of   diffeomorphisms of a torus
  We establish a connection between the strong solution to the spatially periodic Navier-Stokes equations and a solution to a system of forward-backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We construct representations of the strong solution to the Navier-Stokes equations in terms of diffusion processes. 
Ana Bela Cruzeiro and Evelina Shamarova