According to a version of Donsker's theorem, geodesic random walks on Riemannian manifolds converge to the respective Brownian motion. From a computational perspective, however, evaluating geodesics can be quite costly. We therefore introduce approximate geodesic random walks based on the concept of retractions. We show that these approximate walks converge to the correct Brownian motion in the Skorokhod topology as long as the geodesic equation is approximated up to second order. As a result we obtain an efficient algorithm for sampling Brownian motion on compact Riemannian manifolds.

翻译：根据唐斯克理论的版本, 里曼尼方形上的大地学随机行走与相应的布朗运动相汇合。 然而, 从计算角度来说, 评估大地学可能成本相当高。 因此, 我们引入了基于撤回概念的近似大地学随机行走。 我们显示, 这些近似行走与斯科罗克霍德方形学中的正确的布朗运动相汇, 只要大地学方程接近于第二顺序。 因此, 我们获得了一种高效的算法, 用于对紧凑的里曼尼格罗方形上的布朗运动进行取样。