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

Taeyoung Lee, Gregory S. Chirikjian

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.

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

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.
Paper Structure (22 sections, 7 theorems, 80 equations)

This paper contains 22 sections, 7 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Riemannian Manifold
  4. Tensor Fields
  5. Intrinsic Fokker-Planck Equations
  6. Stochastic Processes and the Fokker-Planck Equation
  7. Fokker-Planck Equation for Stratonovich SDE
  8. Fokker-Planck Equation for Itô SDE
  9. Example
  10. Two-Sphere
  11. Riemannian Metric
  12. Orthonormal Frame
  13. Stratonovich SDE
  14. Infinitesimal Generator
  15. Fokker-Planck Equation
  16. ...and 7 more sections

Key Result

Theorem 1

The infinitesimal generator $\mathcal{A}:C^\infty(M)\rightarrow C^\infty(M)$ of the Stratonovich stochastic differential equation eqn:SDE on $f\in C^\infty(M)$ is given by

Theorems & Definitions (18)

  • Theorem 1: Generator of Stratonovich SDE
  • proof
  • Lemma 1
  • proof
  • Theorem 2: Fokker-Planck Equation for Stratonovich SDE
  • proof
  • Example 1: Stratonovich SDE for Brownian Motion
  • Theorem 3: Itô--Stratonovich Conversion
  • proof
  • Theorem 4: Generator of Itô SDE
  • ...and 8 more