The original post is in StackExchange but no one has answered it yet. I personally think it is more related to the research area so I put it in MathOverflow. Below is the question in the original post:

I am looking for any reference that states, and proves, a Fokker-Planck equation for Riemannian manifolds.

In particular, if $dX_t = \mu(X_t) dt + \sigma(X_t)dB_t$ is a stochastic differential equation on a manifold, I want to relate $\mu$ and $\sigma$ to the time evolution of the density of $X_t$, just like the Euclidean Fokker-Planck equation. It would be great if there is a global description of the time evolution, but a local coordinate expression would be okay too.