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 in distribution to the correct Brownian motion 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：根据多恩克尔定理的一种版本，黎曼流形上的测地随机游走收敛于相应的布朗运动。然而，从计算角度来看，计算测地线的成本可能相当高。因此，我们基于回缩概念引入了近似测地随机游走。我们证明了只要测地线方程被近似至二阶精度，这些近似游走在分布上就会收敛到正确的布朗运动。由此，我们得到了一种在紧致黎曼流形上高效采样布朗运动的算法。