Geometric Interpretation of Brownian Motion on Riemannian Manifolds
Taeyoung Lee, Gregory S. Chirikjian
TL;DR
This work develops a unified geometric framework for Brownian motion on nonlinear configuration spaces, deriving stochastic differential equations that realize diffusion with generator $\tfrac{1}{2}\Delta$ on intrinsic manifolds, embedded submanifolds, and Lie groups. By injecting isotropic noise along an orthonormal frame and choosing drift terms that encode geometric data (covariant derivatives of frame fields, mean curvature for embeddings, and the adjoint-trace for Lie groups), the authors obtain explicit Stratonovich and Itô formulations with clear interpretation of curvature-induced drift. A key result is that the Stratonovich drift vanishes for unimodular Lie groups and that, in embedded cases, the Itô drift equals the mean curvature vector; in intrinsic cases the Itô drift vanishes while the Stratonovich drift carries covariant-frame information. The framework is validated through detailed examples (spheres, torus, hyperbolic space, $\mathrm{SO}(3)$, affine group) illustrating how the geometry of each space determines the diffusion dynamics, enabling coordinate-free analysis and numerically stable simulations for nonlinear spaces in mechanics and robotics.
Abstract
This paper presents a unified geometric framework for Brownian motion on manifolds, encompassing intrinsic Riemannian manifolds, embedded submanifolds, and Lie groups. The approach constructs the stochastic differential equation by injecting noise along each axis of an orthonormal frame and designing the drift term so that the resulting generator coincides with the Laplace--Beltrami operator. Both Stratonovich and Itô formulations are derived explicitly, revealing the geometric origin of curvature-induced drift. The drift is shown to correspond to the covariant derivatives of the frame fields for intrinsic manifolds, the mean curvature vector for embedded manifolds, and the adjoint-trace term for Lie groups, which vanishes for unimodular cases. The proposed formulation provides a geometrically transparent and mathematically consistent foundation for diffusion processes on nonlinear configuration spaces.