We define and study the 3-dimensional windings along Brownian paths in the quaternionic Euclidean, projective and hyperbolic spaces. In particular, the asymptotic laws of these windings are shown to be Gaussian for the flat and spherical geometries while the hyperbolic winding exhibits a different long time-behavior. The corresponding asymptotic law seems to be new and is related to the Cauchy relativistic distribution.

中文翻译：

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四元布朗绕组

我们定义并研究了四元数欧几里得空间、射影空间和双曲空间中沿布朗路径的 3 维绕组。特别是，这些绕组的渐近定律对于平面和球形几何图形显示为高斯，而双曲线绕组则表现出不同的长时间行为。相应的渐近定律似乎是新的，并且与柯西相对论分布有关。