Densit\'e des orbites des trajectoires browniennes sous l'action de la transformation de L\'evy

Abstract

Let T be a measurable transformation of a probability space $(E,\mathcal {E},\pi)$, preserving the measure {\pi}. Let X be a random variable with law \pi. Call K(\cdot, \cdot) a regular version of the conditional law of X given T(X). Fix $B\in \mathcal {E}$. We first prove that if B is reachable from \pi-almost every point for a Markov chain of kernel K, then the T-orbit of \pi-almost every point X visits B. We then apply this result to the L\'evy transform, which transforms the Brownian motion W into the Brownian motion |W| - L, where L is the local time at 0 of W. This allows us to get a new proof of Malric's theorem which states that the orbit under the L\'evy transform of almost every path is dense in the Wiener space for the topology of uniform convergence on compact sets.