Forward-backward stochastic differential equations on tensor fields and application to Navier-Stokes equations on Riemannian manifolds

Abstract

In this paper we introduce a class of forward-backward stochastic differential equations on tensor fields of Riemannian manifolds, which are related to semilinear parabolic partial differential equations on tensor fields. Moreover, we will use these forward-backward stochastic differential equations to give a stochastic characterization of incompressible Navier-Stokes equations on Riemannian manifolds, where some extra conditions used in [22] are not required.