Generalized stochastic Lagrangian paths for the Navier-Stokes equation

In the note added in proof of the seminal paper [Groups of diffeomorphisms andthe motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163], Ebinand Marsden introduced the so-called correct Laplacian for the Navier-Stokes equationon a compact Riemannian manifold. In the spirit of Brenier's generalized flows forthe Euler equation, we introduce a class of semimartingales on a compact Riemannianmanifold. We prove that these semimartingales are critical points to the correspondingkinetic energy if and only if its drift term solves weakly the Navier-Stokes equationdefined with Ebin-Marsden's Laplacian. We also show that for the torus case,classical solutions of the Navier-Stokes equation realize the minimum of the kineticenergy in a suitable class.