Lagrangian Navier–Stokes diffusions on manifolds: Variational principle and stability

We prove a variational principle for stochastic flows on manifolds. It extends V.I. Arnoldʼs description of Lagrangian Euler flows, which are geodesics for the L2 metric on the manifold, to the stochastic case. Here we obtain stochastic Lagrangian flows with mean velocity (drift) satisfying the Navier–Stokes equations.We study the stability properties of such trajectories as well as the evolution in time of the rotation between the underlying particles. The case where the underlying manifold is the two-dimensional torus is described in detail.