Random diffeomorphisms and integration of the classical Navier—Stokes equations

We derive random implicit representations for the solutions of the classical Navier—Stokes equations for an incompressible viscous fluid. This program is carried out for Riemannian manifolds (without boundary) which are isometrically embedded in a Euclidean space (spheres, tori, Rn, etc.). Our results appear as an extension to smooth manifolds of the random vortex method of computational fluid dynamics. We derive these representations from gauge-theoretical considerations and the Ito formula for differential forms of stochastic analysis.