The Fokker-Planck equation describes how probability distributions evolve under stochastic dynamics — drift pushes probability, diffusion spreads it. In flat Euclidean space, the equation is a partial differential equation with familiar operators: divergence, gradient, Laplacian. On a curved Riemannian manifold, these operators depend on coordinates, and the standard formulation becomes coordinate-dependent: write the equation in one chart, and it looks different in another.
The paper provides a coordinate-free formulation. In the Stratonovich interpretation, the infinitesimal generator of the diffusion is expressed through Lie derivatives — intrinsic objects that encode how vector fields flow on the manifold without reference to any coordinate system. The adjoint operator, which gives the Fokker-Planck equation itself, follows from the divergence theorem on manifolds. In the Itô interpretation, a diffusion tensor field generalizes the Euclidean diffusion matrix, and the Fokker-Planck equation takes an intrinsic double-divergence form.
The structural point is about what coordinates hide. In Euclidean space, the Fokker-Planck equation looks like it's about derivatives of a function. On a manifold, the same physics — probability transport by drift and diffusion — is about geometric operations on density forms. The divergence is not a calculus operation but a statement about how probability flux flows through surfaces. The Laplacian is not second derivatives but curvature-weighted spreading. The coordinate-free formulation doesn't just generalize the equation — it reveals what the equation always was: a statement about geometry, dressed up as calculus by the accident of flat coordinates.
(arXiv:2603.18320)