Densité des orbites des trajectoires browniennes sous l’action de la transformation de Lévy

Jean Brossard; Christophe Leuridan

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 2, page 477-517
  • ISSN: 0246-0203

Abstract

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Let Tbe a measurable transformation of a probability space ( E , ℰ , π ) , preserving the measureπ. Let X be a random variable with law π. Call K(⋅, ⋅) a regular version of the conditional law of X given T(X). Fix B ∈ ℰ . We first prove that ifB is reachable from π-almost every point for a Markov chain of kernel K, then the T-orbit of π-almost every point X visits B. We then apply this result to the Lévy transform, which transforms the Brownian motion W into the Brownian motion |W| − L, where L is the local time at 0 of W. This allows us to get a new proof of Malric’s theorem which states that the orbit under the Lévy transform of almost every path is dense in the Wiener space for the topology of uniform convergence on compact sets.

How to cite

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Brossard, Jean, and Leuridan, Christophe. "Densité des orbites des trajectoires browniennes sous l’action de la transformation de Lévy." Annales de l'I.H.P. Probabilités et statistiques 48.2 (2012): 477-517. <http://eudml.org/doc/272079>.

@article{Brossard2012,
abstract = {Let Tbe a measurable transformation of a probability space $(E,\mathcal \{E\},\pi )$ , preserving the measureπ. Let X be a random variable with law π. Call K(⋅, ⋅) a regular version of the conditional law of X given T(X). Fix $B\in \mathcal \{E\}$ . We first prove that ifB is reachable from π-almost every point for a Markov chain of kernel K, then the T-orbit of π-almost every point X visits B. We then apply this result to the Lévy transform, which transforms the Brownian motion W into the Brownian motion |W| − L, where L is the local time at 0 of W. This allows us to get a new proof of Malric’s theorem which states that the orbit under the Lévy transform of almost every path is dense in the Wiener space for the topology of uniform convergence on compact sets.},
author = {Brossard, Jean, Leuridan, Christophe},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {brownian motion; Lévy transform; density of orbits; recurrence; ergodicity; Brownian motion; reflected in 0; accessibility; excursion; CUCZ topology},
language = {eng},
number = {2},
pages = {477-517},
publisher = {Gauthier-Villars},
title = {Densité des orbites des trajectoires browniennes sous l’action de la transformation de Lévy},
url = {http://eudml.org/doc/272079},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Brossard, Jean
AU - Leuridan, Christophe
TI - Densité des orbites des trajectoires browniennes sous l’action de la transformation de Lévy
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 2
SP - 477
EP - 517
AB - Let Tbe a measurable transformation of a probability space $(E,\mathcal {E},\pi )$ , preserving the measureπ. Let X be a random variable with law π. Call K(⋅, ⋅) a regular version of the conditional law of X given T(X). Fix $B\in \mathcal {E}$ . We first prove that ifB is reachable from π-almost every point for a Markov chain of kernel K, then the T-orbit of π-almost every point X visits B. We then apply this result to the Lévy transform, which transforms the Brownian motion W into the Brownian motion |W| − L, where L is the local time at 0 of W. This allows us to get a new proof of Malric’s theorem which states that the orbit under the Lévy transform of almost every path is dense in the Wiener space for the topology of uniform convergence on compact sets.
LA - eng
KW - brownian motion; Lévy transform; density of orbits; recurrence; ergodicity; Brownian motion; reflected in 0; accessibility; excursion; CUCZ topology
UR - http://eudml.org/doc/272079
ER -

References

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  1. [1] M. Malric. Densité des zéros des transformés de Lévy itérés d’un mouvement brownien. C. R. Math. Acad. Sci. Paris336 (2003) 499–504. Zbl1024.60034MR1975087
  2. [2] M. Malric. Density of paths of iterated Levy transforms of Brownian motion. Preprint, 2005. Available at arXiv:math/0511154v2. Zbl1274.60171MR2972500
  3. [3] M. Malric. Density of paths of iterated Levy transforms of Brownian motion. Preprint, 2007. Available at arXiv:math/0511154v3. Zbl1274.60171MR2972500
  4. [4] M. Malric. Density of paths of iterated Levy transforms of Brownian motion. Preprint, 2009. Available at arXiv:math/0511154v4. Zbl1274.60171MR2972500
  5. [5] M. Malric. Density of paths of iterated Levy transforms of Brownian motion. ESAIM Probab. Stat. (2012). To appear. DOI:10.1051/ps/2010020, published on line (03 février 2011). Zbl1274.60171MR2972500
  6. [6] K. Petersen. Ergodic Theory. Cambridge Univ. Press, Cambridge, 1989. Corrected reprint of the 1983 original. Zbl0507.28010MR1073173
  7. [7] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1991. Zbl0917.60006MR1083357

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