Global Tensor Field Formulation of the Fokker-Planck Equation on Riemannian Manifolds

TL;DR

This paper develops a coordinate-free, geometric formulation of the Fokker-Planck equation on Riemannian manifolds, enabling intrinsic analysis of diffusion processes in curved spaces.

Contribution

It introduces a global, intrinsic tensor-field approach for the Fokker-Planck equation on Riemannian manifolds, unifying Stratonovich and Ito formulations.

Findings

Provides a coordinate-free geometric form of the Fokker-Planck equation.

Derives an intrinsic double-divergence representation for diffusion.

Offers a globally valid framework for diffusion on curved spaces.

Abstract

This paper presents a global, coordinate-free formulation of the Fokker-Planck equation on Riemannian manifolds. In the Stratonovich formulation, the infinitesimal generator is expressed intrinsically through Lie derivatives, and its adjoint is derived via the divergence theorem, yielding a concise geometric form of the Fokker-Planck equation. In the Ito formulation, a diffusion tensor field is introduced to generalize the Euclidean diffusion matrix, and a tensor-field-based analysis establishes an intrinsic double-divergence representation of the Fokker-Planck equation. The proposed framework provides a globally valid and geometrically consistent interpretation of diffusion and probability transport on Riemannian manifolds, supported by compact and intuitive proofs.