Density of paths of iterated Lévy transforms of brownian motion
ESAIM: Probability and Statistics, Tome 16 (2012) , pp. 399-424.

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 (tBt(n))n⩾0 is dense in Wiener space, for the topology of uniform convergence on compact time intervals.

DOI : https://doi.org/10.1051/ps/2011020
Classification : 60g99,  60j65,  37a05,  37a50,  37a25
Mots clés : brownian motion, Lévy transform, excursions, zeroes of brownian motion, ergodicity
@article{PS_2012__16__399_0,
     author = {Malric, Marc},
     title = {Density of paths of iterated L\'evy transforms of brownian motion},
     journal = {ESAIM: Probability and Statistics},
     pages = {399--424},
     publisher = {EDP-Sciences},
     volume = {16},
     year = {2012},
     doi = {10.1051/ps/2011020},
     zbl = {1274.60171},
     mrnumber = {2972500},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2011020/}
}
Malric, Marc. Density of paths of iterated Lévy transforms of brownian motion. ESAIM: Probability and Statistics, Tome 16 (2012) , pp. 399-424. doi : 10.1051/ps/2011020. http://archive.numdam.org/articles/10.1051/ps/2011020/

[1] L.E. Dubins and M. Smorodinsky, The modified, discrete Lévy transformation is Bernoulli, in Séminaire de Probabilités XXVI. Lect. Notes Math. 1526 (1992) | Numdam | Zbl 0761.60043

[2] M. Malric, Densité des zéros des transformées de Lévy itérées d'un mouvement brownien. C. R. Acad. Sci. Paris, Sér. I 336 (2003) 499-504. | MR 1975087 | Zbl 1024.60034

[3] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3th edition. Springer-Verlag, Berlin (1999) | MR 1725357 | Zbl 0731.60002