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

TL;DR

This paper investigates the density of orbits of Brownian motion paths under the Lévy transform, providing a new proof that almost all paths are dense in the Wiener space with respect to uniform convergence.

Contribution

It establishes a general criterion for orbit density under measure-preserving transformations and applies it to the Lévy transform, offering a novel proof of Malric's theorem.

Findings

Almost every Brownian path has a dense orbit under the Lévy transform.

The new proof simplifies understanding of orbit density in Wiener space.

The approach links Markov chain reachability with orbit properties.

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.