Geometric Interpretation of Brownian Motion on Riemannian Manifolds

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 It\^{o} 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.