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

Taeyoung Lee; Gregory S. Chirikjian

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arXiv · 2603.18320

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

Taeyoung Lee·Gregory S. Chirikjian

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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.

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BibTeX RIS

Taeyoung Lee, Gregory S. Chirikjian. 2026-03-18. Global Tensor Field Formulation of the Fokker-Planck Equation on Riemannian Manifolds. https://arxiv.org/abs/2603.18320

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