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

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.

### 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
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