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

TL;DR

This paper introduces forward-backward stochastic differential equations on tensor fields of Riemannian manifolds and uses them to provide a stochastic characterization of Navier-Stokes equations on such manifolds, simplifying previous conditions.

Contribution

It develops a new class of stochastic differential equations on tensor fields and applies them to characterize Navier-Stokes equations without extra assumptions.

Findings

Stochastic representation of Navier-Stokes equations on Riemannian manifolds

Elimination of certain conditions previously required

New framework for stochastic analysis on tensor fields

Abstract

In this paper we introduce a class of forward-backward stochastic differential equations on tensor fields of Riemannian manifolds, which are related to semi-linear 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.