STOCHASTIC GEODESICS AND FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS ON LIE GROUPS

摘要

We describe how to generalize to the stochastic case the notion of geodesic on a Lie group equipped with an invariant metric. As second order equations (in time), stochastic geodesics are characterized in terms of stochastic forward-backward differential systems. When the group is the diffeomorphisms group this corresponds to a probabilistic description of the Navier-Stokes equations.