Density of paths of iterated Lévy transforms of Brownian motion. (English) Zbl 1274.60171

Malric, Marc

ESAIM, Probab. Stat. 16, 399-424 (2012).

The Lévy transform is the transformation \(T\) of the Wiener space given by \(T(B)(t)=\int_0^t\text{sgn}(B(s))\,dB(s)\), \(t\geq0\). The Wiener measure is invariant under \(T\). In the present contribution it is proved that \(T\) is topologically recurrent, i.e. for almost every element \(w\) of the Wiener space the orbit \(\{T^nw:n\geq0\}\) is dense in the Wiener space (with respect to the topology of uniform convergence on compact intervals). While this is necessary for ergodicity of \(T\) with respect to the Wiener measure, ergodicity remains an open question.

Reviewer: Hans Crauel (Frankfurt am Main)

MSC:

60G99	Stochastic processes
37A05	Dynamical aspects of measure-preserving transformations
37A25	Ergodicity, mixing, rates of mixing
37A50	Dynamical systems and their relations with probability theory and stochastic processes
60J65	Brownian motion

Keywords:

Lévy transform; topological recurrence; ergodicity