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Density of paths of iterated Lévy transforms of Brownianmotion

Published online by Cambridge University Press:  31 August 2012

Marc Malric
Marc Malric*
Affiliation:
15, avenue Gambetta, 94160 St-Mandé, France. marc.malric@gmail.com
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Abstract

The Lévy transform of a Brownian motion B is the Brownian motionB(1) given by Bt(1) = ∫0tsgn(Bs)dBs; callB(n) the Brownian motion obtained fromB by iterating n times this transformation. Weestablish that almost surely, the sequence of paths (t → Bt(n))n⩾0isdense in Wiener space, for the topology of uniform convergence on compact timeintervals.

Keywords

Brownian motionLévy transformexcursionszeroes of Brownian motionergodicity

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 399 - 424
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

Dubins, L.E. and Smorodinsky, M., The modified, discrete Lévy transformation is Bernoulli, in Séminaire de Probabilités XXVI. Lect. Notes Math. 1526 (1992) CrossRefGoogle Scholar
Malric, M., Densité des zéros des transformées de Lévy itérées d’un mouvement brownien. C. R. Acad. Sci. Paris, Sér. I 336 (2003) 499–504. Google Scholar
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3th edition. Springer-Verlag, Berlin (1999)