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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.

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An early reference is Coordinate-independent formulation of the Langevin equation (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.

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