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Density of paths of iterated Lévy transforms of Brownian motion

Published online by Cambridge University Press:  31 August 2012

Marc Malric
Marc Malric*
Affiliation:
15, avenue Gambetta, 94160 St-Mandé, France. [email protected]
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Abstract

The Lévy transform of a Brownian motion B is the Brownian motion B(1) given by Bt(1) = ∫0tsgn(Bs)dBs; call B(n) the Brownian motion obtained from B by iterating n times this transformation. We establish that almost surely, the sequence of paths (t → Bt(n))n⩾0 is dense in Wiener space, for the topology of uniform convergence on compact time intervals.

Keywords

Brownian motionLévy transformexcursionszeroes of Brownian motionergodicity
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)