A note on Constantin and Iyer's representation formula for the Navier-Stokes equations

The purpose of this note is to establish a probabilistic representation formula for the Navier-Stokes equations on compact Riemannian manifolds. Such a formula has been provided by Constantin and Iyer in the flat case of $\mathbb R^n$ or of $\mathbb T^n$. On a Riemannian manifold, however, there are several different choices of Laplacian operators acting on vector fields. In this paper, we shall use the de Rham-Hodge Laplacian operator which seems more relevent to the probabilistic setting, and adopt Elworthy-Le Jan-Li's idea to decompose it as a sum of the square of Lie derivatives.