Quaternionic Brownian Windings
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Quaternionic Brownian Windings Fabrice Baudoin1 · Nizar Demni2 · Jing Wang3 Received: 8 May 2020 / Revised: 22 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We define and study the three-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. Keywords Stochastic winding · Large time asymptotic · Quaternionic projective space · Quaternionic hyperbolic space · Cauchy relativistic distribution Mathematics Subject Classification 58J65 · 53C26 · 60J60
1 Introduction In the punctured complex plane C\{0}, consider the one-form α=
xdy − ydx . x 2 + y2
Author supported in part by the Simons Foundation and NSF Grant DMS-1901315. Author supported by the NSF Grant DMS-1855523.
B
Jing Wang [email protected] Fabrice Baudoin [email protected] Nizar Demni [email protected]
1
Department of Mathematics, University of Connecticut, Storrs CT 06269, USA
2
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
3
Department of Mathematics, Department of Statisctics, Purdue University, West Lafayette IN 47907, USA
123
Journal of Theoretical Probability
For every smooth path γ : [0, +∞) → C\{0} one has the representation γ (t) = |γ (t)| exp i
γ [0,t]
α , t ≥ 0.
It is therefore natural to call α the winding form around 0 since the integral of a smooth path γ along this form quantifies the angular motion of this path. The integral of the winding form along the paths of a two-dimensional Brownian motion (B(t))t≥0 which is not started from 0 can be defined using Itô’s calculus and yields the Brownian winding functional: α.
ζ (t) = B[0,t]
This functional and several natural variations of it have been extensively studied in the literature. In particular, the famous Spitzer’s theorem states that in distribution, when t → +∞ the following convergence takes place in distribution 2 ζ (t) → C1 ln t where C1 is a Cauchy distribution with parameter 1. We refer the reader to [10] and [6] and references therein for more details about the Brownian winding functional. More recently, Brownian winding functionals were studied in the paper [1] in the complex projective space and the complex hyperbolic space. Our goal in this paper is to introduce a natural generalization of the winding form in homogeneous four-dimensional spaces equipped with a quaternionic structure and study the limiting laws of the integrals of this form along the corresponding Brownian motion paths. Unlike the complex case studied in [1], one can not make use of the theory of analytic functions. Actually, it turns out that the quaternionic winding form is valued in the three-dimensional Lie algebra su(2) and quantifies in a natural way the a
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