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.


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.