Density of orbits of Brownian trajectories under the action of the Lévy transformation
type: Item
instance of: scholarly article
title: Density of orbits of Brownian trajectories under the action of the Lévy transformation
zbMATH Open document ID: 1248.37014
DOI: 10.1214/11-AIHP463
author: Jean Brossard
author: Christophe Leuridan
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2012-06-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1208.0151
full work available at URL: https://projecteuclid.org/euclid.aihp/1334148208
Mathematics Subject Classification ID: 37A50
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 6042936
zbMATH Keywords: Brownian motion
zbMATH Keywords: reflected in \(0\)
zbMATH Keywords: density of orbits
zbMATH Keywords: accessibility
zbMATH Keywords: excursion
zbMATH Keywords: Lévy transform
zbMATH Keywords: CUCZ topology
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
arXiv ID: 1208.0151
cites work: Densité des zéros des transformés de Lévy itérés d'un mouvement brownien. (Density of zeros for iterated Lévy transforms of Brownian motion)
cites work: Density of paths of iterated Lévy transforms of Brownian motion
cites work: Q3993280
cites work: Q3997782
OpenAlex ID: W3101008189
review text: Let \(W\) be the space of all continuous maps \(w:[0,\infty )\rightarrow {\mathbb R}\), \(w(0)=0\), with the Wiener measure \(P\). The Lévy transform \(W\rightarrow W\) is defined as taking \(w\) into \(|w(t)|-l(t,\omega )\), where \(l(t,w)=\liminf_{\varepsilon \rightarrow 0}(2\varepsilon )^{- 1}\int_{_{0}}^{^{t}}1_{(|w(s)|\leq \varepsilon )}ds\) is the local time in \(0\). It preserves the Wiener measure. It is an unsolved problem whether the Lévy transform is ergodic or not. Recently, \textit{M. Malric} succeeded in making a step towards its solution by proving the density, in the uniform convergence on compacts, of the iterates of this transform for almost all \(w\) [``Density of paths of iterated Lévy transforms of Brownian motion'', ESAIM, Probab. Stat. 16, 399--424 (2012; \url{doi:10.1051/ps/2011020}), \url{arXiv: math/0511154}].NEWLINENEWLINE The authors present in this paper a detailed alternative demonstration of this result (Theorem 1). The paper starts with the following general result: if \(T\) is a map preserving a probability \(\pi\) (on a space \(E\)), \(X\) is a \(\pi\)-distributed random variable and \(K\) is a conditional distribution of \(X\) with respect to \(T(X)\), then NEWLINE\[NEWLINE\bigcup_{n\in {\mathbb N}}(K^{n}(\cdot ,B)>0)\subset \bigcup_{n\in {\mathbb N}}T^{-n}(B)=\limsup_{n\rightarrow +\infty }T^{-n}(B) \text{ a.e. }NEWLINE\]NEWLINE If \(E\) is topological with a denumerable basis, then, in order that the orbit by \(T\) of \(\pi\)-almost every point in \(E\) is dense, it is sufficient that every open set \(V\subset E\) is accessible from almost every point by the \(K\)-Markov chain. Consider the set \(Q\) of positive rationals ordered lexicographically by \((a+b,a)\) in the irreducible writing \(a/b\) and denote, for an interval \(A\subset(0,\infty)\), by \(q(A)\) the smallest \(q\in A\cap Q\) in the order in \(Q\). For \(r\in W_{+}=\{w;\in W,w\geq 0\}\) and for \(\varepsilon \in E=\{-1,1\}^{Q}\), \(t>0\) let \(I=(a,b)\) be the interv…
description: scientific article; zbMATH DE number 6042936