Density of paths of iterated Lévy transforms of Brownian motion

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Received: 24 February 2009

Abstract

The Lévy transform of a Brownian motion B is the Brownian motion B⁽¹⁾ given by B_t⁽¹⁾ = ∫₀^tsgn(B_s)dB_s; 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 → B_t⁽ⁿ⁾)_n⩾0 is dense in Wiener space, for the topology of uniform convergence on compact time intervals.