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Windings of Planar Stable Processes

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Séminaire de Probabilités XLV

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2078))

Abstract

Using a generalization of the skew-product representation of planar Brownian motion and the analogue of Spitzer’s celebrated asymptotic Theorem for stable processes due to Bertoin and Werner, for which we provide a new easy proof, we obtain some limit Theorems for the exit time from a cone of stable processes of index α ∈ (0, 2). We also study the case t → 0 and we prove some Laws of the Iterated Logarithm (LIL) for the (well-defined) winding process associated to our planar stable process.

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Notes

  1. 1.

    When we simply write: Brownian motion, we always mean real-valued Brownian motion, starting from 0. For two-dimensional Brownian motion, we indicate planar or complex BM.

References

  1. F. Aurzada, L. Döring, M. Savov, Small time Chung type LIL for Lévy processes. Bernoulli 19, 115–136 (2013)

    Article  MATH  Google Scholar 

  2. R. Banũelos, K. Bogdan, Symmetric stable processes in cones. Potential Anal. 21, 263–288 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Bertoin, Lévy Processes (Cambridge University Press, Cambridge, 1996)

    MATH  Google Scholar 

  4. J. Bertoin, R.A. Doney, Spitzer’s condition for random walks and Lévy processes. Ann. Inst. Henri Poincaré 33, 167–178 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Bertoin, W. Werner, Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process, in Séminaire de Probabilités XXVIII, ed. by J. Azéma, M. Yor, P.A. Meyer. Lecture Notes in Mathematics, vol. 1583 (Springer, Berlin, 1994), pp. 138–152

    Google Scholar 

  6. J. Bertoin, W. Werner, Compertement asymptotique du nombre de tours effectués par la trajectoire brownienne plane. Séminaire de Probabilités XXVIII, ed. by J. Azéma, M. Yor, P.A. Meyer. Lecture Notes in Mathematics, vol. 1583 (Springer, Berlin, 1994), pp. 164–171

    Google Scholar 

  7. J. Bertoin, W. Werner, Stable windings. Ann. Probab. 24, 1269–1279 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Breiman, A delicate law of the iterated logarithm for non-decreasing stable processes. Ann. Math. Stat. 39, 1818–1824 (1968); correction 41, 1126

    Google Scholar 

  9. D. Burkholder, Exit times of Brownian Motion, harmonic majorization and hardy spaces. Adv. Math. 26, 182–205 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  10. M.E. Caballero, J.C. Pardo, J.L. Pérez, Explicit identities for Lvy processes associated to symmetric stable processes. Bernoulli 17(1), 34–59 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. O. Chybiryakov, The Lamperti correspondence extended to Lévy processes and semi-stable Markov processes in locally compact groups. Stoch. Process. Appl. 116, 857–872 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. R.D. De Blassie, The first exit time of a two-dimensional symmetric stable process from a wedge. Ann. Probab. 18, 1034–1070 (1990)

    Article  MathSciNet  Google Scholar 

  13. R.A. Doney, Small time behaviour of Lévy processes. Electron. J. Probab. 9, 209–229 (2004)

    Article  MathSciNet  Google Scholar 

  14. R.A. Doney, R.A. Maller, Random walks crossing curved boundaries: a functional limit theorem, stability and asymptotic distributions for exit positions. Adv. Appl. Probab. 32, 1117–1149 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. R.A. Doney, R.A. Maller, Stability of the overshoot for Lévy processes. Ann. Probab. 30, 188–212 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. R.A. Doney, R.A. Maller, Stability and attraction to normality for Le’vy processes at zero and infinity. J. Theor. Probab. 15, 751–792 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Durrett, A new proof of Spitzer’s result on the winding of 2-dimensional Brownian motion. Ann. Probab. 10, 244–246 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. B.E. Fristedt, The behavior of increasing stable processes for both small and large times. J. Math. Mech. 13, 849–856 (1964)

    MathSciNet  MATH  Google Scholar 

  19. B.E. Fristedt, Sample function behavior of increasing processes with stationary, independent increments. Pacific J. Math. 21(1), 21–33 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  20. S.E. Graversen, J. Vuolle-Apiala, α-Self-similar Markov processes. Probab. Theor. Relat. Field. 71, 149–158 (1986)

    Google Scholar 

  21. K. Itô, H.P. McKean, Diffusion Processes and their Sample Paths (Springer, Berlin, 1965)

    Book  MATH  Google Scholar 

  22. J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes, 2nd edn. (Springer, Berlin, 2003)

    Book  MATH  Google Scholar 

  23. A. Khintchine, Sur la croissance locale des processus stochastiques homogènes à accroissements indépendants. (Russian article and French resume) Akad. Nauk. SSSR Izv. Ser. Math. 3(5–6), 487–508 (1939)

    Google Scholar 

  24. S.W. Kiu, Semi-stable Markov processes in \({\mathbb{R}}^{n}\). Stoch. Process. Appl. 10(2), 183–191 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  25. A.E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications (Springer, Berlin, 2006)

    MATH  Google Scholar 

  26. J. Lamperti, Semi-stable Markov processes I. Z. Wahr. Verw. Gebiete, 22, 205–225 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  27. J.F. Le Gall, Some properties of planar Brownian motion, in Cours de l’école d’été de St-Flour XX, ed. by A. Dold, B. Eckmann, E. Takens. Lecture Notes in Mathematics, vol. 1527 (Springer, Berlin, 1992), pp. 111–235

    Google Scholar 

  28. J.F. Le Gall, M. Yor, Etude asymptotique de certains mouvements browniens complexes avec drift. Probab. Theor. Relat. Field. 71(2), 183–229 (1986)

    Article  MATH  Google Scholar 

  29. J.F. Le Gall, M. Yor, Etude asymptotique des enlacements du mouvement brownien autour des droites de l’espace. Probab. Theor. Relat. Field. 74(4), 617–635 (1987)

    Article  MATH  Google Scholar 

  30. M. Liao, L. Wang, Isotropic self-similar Markov processes. Stoch. Process. Appl. 121(9), 2064–2071 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. P. Messulam, M. Yor, On D. Williams’ “pinching method” and some applications. J. Lond. Math. Soc. 26, 348–364 (1982)

    Google Scholar 

  32. J.W. Pitman, M. Yor, The asymptotic joint distribution of windings of planar Brownian motion. Bull. Am. Math. Soc. 10, 109–111 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  33. J.W. Pitman, M. Yor, Asymptotic laws of planar Brownian motion. Ann. Probab. 14, 733–779 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  34. J.W. Pitman, M. Yor, Further asymptotic laws of planar Brownian motion. Ann. Probab. 17(3), 965–1011 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  35. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)

    Book  MATH  Google Scholar 

  36. Z. Shi, Liminf behaviours of the windings and Lévy’s stochastic areas of planar Brownian motion. Séminaire de Probabilités XXVIII, ed. by J. Azéma, M. Yor, P.A. Meyer. Lecture Notes in Mathematics, vol. 1583 (Springer, Berlin, 1994), pp. 122–137

    Google Scholar 

  37. Z. Shi, Windings of Brownian motion and random walks in the plane. Ann. Probab. 26(1), 112–131 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  38. A.V. Skorohod, Random Processes with Independent Increments (Kluwer, Dordrecht, 1991)

    Book  MATH  Google Scholar 

  39. F. Spitzer, Some theorems concerning two-dimensional Brownian motion. Trans. Am. Math. Soc. 87, 187–197 (1958)

    MathSciNet  MATH  Google Scholar 

  40. S. Vakeroudis, Nombres de tours de certains processus stochastiques plans et applications à la rotation d’un polymère. (Windings of some planar Stochastic Processes and applications to the rotation of a polymer). Ph.D. Dissertation, Université Pierre et Marie Curie (Paris VI), April 2011.

    Google Scholar 

  41. S. Vakeroudis, On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol’s identity. SIAM Theor. Probab. Appl. 56(3), 485–507 (2012) [originally published in Teor. Veroyatnost. i Primenen. 56(3), 566–591 (2011)]

    Google Scholar 

  42. S. Vakeroudis, M. Yor, Integrability properties and limit theorems for the exit time from a cone of planar Brownian motion. arXiv preprint arXiv:1201.2716 (2012), to appear in Bernoulli

    Google Scholar 

  43. W. Whitt, Some useful functions for functional limit theorems. Math. Oper. Res. 5, 67–85 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  44. D. Williams, A simple geometric proof of Spitzer’s winding number formula for 2-dimensional Brownian motion. University College, Swansea. Unpublished, 1974

    Google Scholar 

  45. M. Yor, Loi de l’indice du lacet Brownien et Distribution de Hartman-Watson. Z. Wahrsch. verw. Gebiete 53, 71–95 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Yor, Generalized meanders as limits of weighted Bessel processes, and an elementary proof of Spitzer’s asymptotic result on Brownian windings. Studia Scient. Math. Hung. 33, 339–343 (1997)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author S. Vakeroudis is very grateful to Prof. M. Yor for the financial support during his stay at the University of Manchester as a Post Doc fellow invited by Prof. R.A. Doney.

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Authors and Affiliations

  1. Probability and Statistics Group, School of Mathematics, University of Manchester, Alan Turing Building, Oxford Road, Manchester, M13 9PL, UK

    R. A. Doney & S. Vakeroudis

  2. Laboratoire de Probabilités et Modèles Aléatoires (LPMA) CNRS: UMR7599, Université Pierre et Marie Curie - Paris VI, Université Paris-Diderot - Paris VII, 4 Place Jussieu, 75252, Paris Cedex 05, France

    S. Vakeroudis

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  1. R. A. Doney
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Correspondence to S. Vakeroudis .

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Editors and Affiliations

  1. , Laboratoire de Mathématiques, Université de Versailles-St Quentin, avenue des États-Unis 45, Versailles, 78035, France

    Catherine Donati-Martin

  2. IECN, Campus scientifique, BP 239, Nancy-Université, INRIA, Vandoeuvre-lès-Nancy, 54506, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des Etats-Unis 45, Versailles, 78035, France

    Alain Rouault

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Doney, R.A., Vakeroudis, S. (2013). Windings of Planar Stable Processes. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_10

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