论文标题
诱导扭转的随机谎言括号：嘈杂矢量场的模型
Random Lie Brackets that Induce Torsion: A Model for Noisy Vector Fields
论文作者
论文摘要
我们定义并研究一个随机的谎言括号，以引起预期的扭转。几乎所有对歧管的随机分析都假定平行运输。从数学上讲，这个假设是非常合理的。但是，在许多应用的几何形状和图形问题中没有达到平行传输，“坐标的变化”并不是由于噪声而确切的。我们在歧管上制定了一个随机模型，该模型平行运输无法保持并分析该模型在Riemannian几何形状中研究的经典数量方面的后果。我们首先定义一个随机的谎言支架，该括号诱导随机协变量的衍生物。然后，我们研究随机协变量衍生物所隐含的联系，并注意随机架托架会诱导扭转。然后，我们指出诱导的随机测量方程和平行运输的随机微分方程。我们还为我们的构造和随机的拉普拉斯 - 贝特拉米操作员得出了曲率张量。我们讨论了建筑的动机和相关性。
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.
