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Quaternionic Brownian Windings

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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.

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References

  1. Baudoin, F., Wang, J.: Stochastic areas, winding numbers and Hopf fibrations. Probab. Theory Related Fields 169(3–4), 977–1005 (2017)

    Article  MathSciNet  Google Scholar 

  2. Byczkowski, M.T., Malecki, J., Ryznar, M.: Bessel potentials, Hitting distributions, and Green functions. Trans. AMS 361(9), 4871–4900 (2009)

    Article  MathSciNet  Google Scholar 

  3. Chen, K.T.: Iterated integrals and exponential homomorphisms. Proc. London Math. Soc. 4(3), 502–512 (1954)

    Article  MathSciNet  Google Scholar 

  4. Demni, N.: Generalized Stochastic areas and windings arising from Anti-de Sitter and Hopf fibrations. Submitted

  5. Pauwels, E.J., Rogers, L.C.G.: Skew-product decompositions of Brownian motions. Contemp. Math. 73, 237–262 (1988)

    Article  MathSciNet  Google Scholar 

  6. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Third Edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1999)

    Google Scholar 

  7. Volchkov, V., Volchkov, V.: Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg group. Springer Science & Business Media, Berlin (2009)

    Book  Google Scholar 

  8. Watanabe, S.: Asymptotic windings of Brownian motion paths on Riemann surfaces. Acta Appl. Math. 63, 441–464 (2000)

    Article  MathSciNet  Google Scholar 

  9. Yor, M.: Remarques sur une formule de Paul Lévy. Sém. Proba. XIV. (Lecture Notes in Mathematics, vol. 784.) Springer, Berlin, pp. 343-346 (1980)

  10. Yor, M.: Loi de l’indice du lacet Brownien, et distribution de Hartman–Watson. Z. Wahrsch. Verw. Gebiete 53(1), 71–95 (1980)

    Article  MathSciNet  Google Scholar 

  11. Yor, M.: Some Aspects of Brownian Motion. Part I. Some Special Functionals. Lectures in Mathematics ETH Zürich. Birkhuser Verlag, Basel (1992)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA

    Fabrice Baudoin

  2. IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042, Rennes cedex, France

    Nizar Demni

  3. Department of Mathematics, Department of Statisctics, Purdue University, West Lafayette, IN 47907, USA

    Jing Wang

Authors
  1. Fabrice Baudoin
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  2. Nizar Demni
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  3. Jing Wang
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Correspondence to Jing Wang.

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Author supported in part by the Simons Foundation and NSF Grant DMS-1901315.

Author supported by the NSF Grant DMS-1855523.

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Baudoin, F., Demni, N. & Wang, J. Quaternionic Brownian Windings. J Theor Probab 34, 2368–2385 (2021). https://doi.org/10.1007/s10959-020-01034-9

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  • DOI: https://doi.org/10.1007/s10959-020-01034-9

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