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.
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}, mrnumber = {2972500}, zbl = {1274.60171}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps/2011020/} }
TY - JOUR AU - Malric, Marc TI - Density of paths of iterated Lévy transforms of brownian motion JO - ESAIM: Probability and Statistics PY - 2012 SP - 399 EP - 424 VL - 16 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2011020/ DO - 10.1051/ps/2011020 LA - en ID - PS_2012__16__399_0 ER -
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://www.numdam.org/articles/10.1051/ps/2011020/
[1] The modified, discrete Lévy transformation is Bernoulli, in Séminaire de Probabilités XXVI. Lect. Notes Math. 1526 (1992) | Numdam | Zbl
and ,[2] Densité des zéros des transformées de Lévy itérées d'un mouvement brownien. C. R. Acad. Sci. Paris, Sér. I 336 (2003) 499-504. | MR | Zbl
,[3] Continuous Martingales and Brownian Motion, 3th edition. Springer-Verlag, Berlin (1999) | MR | Zbl
and ,Cité par Sources :