The covariant Langevin equation of diffusion on Riemannian manifolds

TL;DR

This paper derives a covariant Langevin equation for diffusion on Riemannian manifolds, ensuring it aligns with the covariant Fokker-Planck equation by imposing a divergence-free frame condition.

Contribution

It introduces the first covariant Langevin stochastic differential equation on Riemannian manifolds with a divergence-free frame constraint.

Findings

Derivation of a covariant Langevin equation using the vielbein formalism

Establishment of a divergence-free frame condition for covariance

Connection between the Langevin SDE and the covariant Fokker-Planck equation

Abstract

The covariant form of the multivariable diffusion-drift process is described by the covariant Fokker--Planck equation using the standard toolbox of Riemann geometry. The covariant form of the equivalent Langevin stochastic differential equation is long sought after in both physics and mathematics. We show that the simplest covariant Stratonovich stochastic differential equation depending on the local orthogonal frame (cf. vielbein) becomes the desired covariant Langevin equation provided we impose an additional covariant constraint: the vectors of the frame must be divergence-free.