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.

中文翻译：

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黎曼流形上的高效随机游走

根据唐斯克定理的一个版本，黎曼流形上的测地随机游走收敛于相应的布朗运动。然而，从计算的角度来看，评估测地线的成本可能相当高。因此，我们引入基于回缩概念的近似测地随机游走。我们证明，只要测地线方程近似到二阶，这些近似行走就会在分布上收敛到正确的布朗运动。因此，我们获得了一种在紧凑黎曼流形上采样布朗运动的有效算法。