The [original post](https://math.stackexchange.com/questions/4158432/reference-request-for-a-riemannian-fokker-planck-equation) 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.



Reference request for a Riemannian Fokker-Planck equation
An early reference is [Coordinate-independent formulation of the Langevin equation](https://doi.org/10.1063/1.527396) (1986).



> 
> A diffusion process on a compact Riemannian manifold is considered,
> and a coordinate-invariant Fokker-Planck equation is formulated. A
> covariant form of the Langevin equation is also derived, and the
> formalism is applied to the stochastic quantization of lattice gauge
> theories.
> 
> 
> 


