Density of paths of iterated Lévy transforms of Brownian motion
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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.
Mathematics Subject Classification: 60g99 / 60j65 / 37a05 / 37a50 / 37a25
Key words: Brownian motion / Lévy transform / excursions / zeroes of Brownian motion / ergodicity
© EDP Sciences, SMAI, 2012