Abstract
We propose a possible way of attacking the question posed originally by Daniel Revuz and Marc Yor in their book published in 1991. They asked whether the Lévy transformation of the Wiener-space is ergodic. Our main results are formulated in terms of a strongly stationary sequence of random variables obtained by evaluating the iterated paths at time one. Roughly speaking, this sequence has to approach zero “sufficiently fast”. For example, one of our results states that if the expected hitting time of small neighborhoods of the origin do not grow faster than the inverse of the size of these sets then the Lévy transformation is strongly mixing, hence ergodic.
The European Union and the European Social Fund have provided financial support to the project under the grant agreement no. TÁMOP 4.2.1./B-09/1/KMR-2010-0003.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968)
J. Brossard, C. Leuridan, Densité des orbites des trajectoires browniennes sous l’action de la transformation de Lévy. Ann. Inst. H. Poincaré Probab. Stat. 48(2), 477–517 (2012). doi:10.1214/11-AIHP463
L.E. Dubins, M. Smorodinsky, The modified, discrete, Lévy-transformation is Bernoulli. In Séminaire de Probabilités, XXVI, Lecture Notes in Math., vol. 1526 (Springer, Berlin, 1992), pp. 157–161
L.E. Dubins, M. Émery, M. Yor, On the Lévy transformation of Brownian motions and continuous martingales. In Séminaire de Probabilités, XXVII, Lecture Notes in Math., vol. 1557 (Springer, Berlin, 1993), pp. 122–132
T. Fujita, A random walk analogue of Lévy’s theorem. Studia Sci. Math. Hungar. 45(2), 223–233 (2008). doi:10.1556/SScMath.45.2008.2.50
J. Jacod, A.N. Shiryaev, Limit theorems for stochastic processes, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288 (Springer, Berlin, 1987)
M. Malric, Transformation de Lévy et zéros du mouvement brownien. Probab. Theory Relat. Fields 101(2), 227–236 (1995). doi:10.1007/BF01375826
M. Malric, Densité des zéros des transformés de Lévy itérés d’un mouvement brownien. C. R. Math. Acad. Sci. Paris 336(6), 499–504 (2003)
M. Malric, Density of paths of iterated Lévy transforms of Brownian motion. ESAIM Probab. Stat. 16, 399–424 (2012). doi:10.1051/ps/2010020
T.F. Móri, G.J. Székely, On the Erdős-Rényi generalization of the Borel-Cantelli lemma. Studia Sci. Math. Hungar. 18(2–4), 173–182 (1983)
D. Revuz, M. Yor, Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293. (Springer, Berlin, 1991)
F. Riesz, B. Sz.-Nagy, Functional Analysis (Frederick Ungar Publishing, New York, 1955). Translated by Leo F. Boron
Acknowledgements
Crucial part of this work was done while visiting the University of Strasbourg, in February of 2011. I am very grateful to IRMA and especially to professor Michel Émery for their invitation and for their hospitality.
Conversations with Christophe Leuridan and Jean Brossard, and a few days later with Marc Yor and Marc Malric inspired the first formulation of Theorems 7 and 8.
The author is grateful to the anonymous referee for his detailed reports and suggestions that improved the presentation of the results significantly. Especially, the referee proposed a simpler argument for stronger result in Theorem 11, pointed out a sloppiness in the proof of Theorem 8, suggested a better formulation of Theorems 7 and 8 and one of his/her remarks motivated Theorem 9.
The original proof of Corollary 20 was based on the Erdős–Rényi generalization of the Borel–Cantelli lemma, see [10]. The somewhat shorter proof in the text was proposed by the referee.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Prokaj, V. (2013). Some Sufficient Conditions for the Ergodicity of the Lévy Transformation. 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_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-00321-4_2
Published:
Publisher Name: Springer, Heidelberg
Print ISBN: 978-3-319-00320-7
Online ISBN: 978-3-319-00321-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)