I have a bit of an exotic partial differential equation:
Let $(M, g)$ be a Riemannian Manifold, $D^{\mu \nu}$ a positive definite symmetric 2-tensor and $\mathfrak{F_{\mu}}$ a covariant tensor on $M.$
We consider the partial differential equation
$$\frac{\partial n}{\partial t} = \nabla_{\mu} D^{\mu \nu}(\nabla_{\nu}-\mathfrak{F}_{\nu})n.$$
I would now like to derive its Ito process and also solve the equation with the Feynman-Kac formula.
Can anyone help me? I was looking for literature, but all the generalizations for manifolds only did standard Brownian motion without a vector field as a drift. So I was wondering if somebody might have a reference.
Thanks in advance!