诱导挠率的随机李括号：噪声向量场的一个模型
Random Lie Brackets that Induce Torsion: A Model for Noisy Vector Fields
摘要：
定义并研究了一个在期望中诱导挠率的随机李括号。几乎所有流形上的随机分析都假设平行输运。从数学上讲，这个假设是非常合理的。然而，在许多应用几何和图形问题中，并行传输是无法实现的，由于噪声的影响，“坐标变换”并不精确。我们制定了一个随机模型的流形上的平行运输不成立，并分析了这个模型的后果，在黎曼几何研究的经典数量。我们首先定义了一个随机李括号，诱导随机共变导数。然后，我们研究的随机共变导数隐含的连接，并注意到，随机李括号诱导扭转。然后，我们陈述诱导随机测地线方程和一个随机微分方程的平行运输。我们还推导出我们的建设和随机Laplace-Beltrami算子的曲率张量。最后，我们讨论了我们的建设的动机和相关性。
英文摘要：
We define and study a random Lie bracket that induces torsion in expectation. Almost all stochastic analysis on manifolds have assumed parallel transport. Mathematically this assumption is very reasonable. However, in many applied geometry and graphics problems parallel transport is not achieved, the "change in coordinates" are not exact due to noise. We formulate a stochastic model on a manifold for which parallel transport does not hold and analyze the consequences of this model with respect to classic quantities studied in Riemannian geometry. We first define a stochastic lie bracket that induces a stochastic covariant derivative. We then study the connection implied by the stochastic covariant derivative and note that the stochastic lie bracket induces torsion. We then state the induced stochastic geodesic equations and a stochastic differential equation for parallel transport. We also derive the curvature tensors for our construction and a stochastic Laplace-Beltrami operator. We close with a discussion of the motivation and relevance of our construction.
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