Some sufficient conditions for the ergodicity of the L\'evy transformation

TL;DR

This paper investigates conditions under which the Lévy transformation of Wiener space is ergodic, focusing on the behavior of a stationary sequence derived from iterated paths and their hitting times.

Contribution

It provides new sufficient conditions for ergodicity of the Lévy transformation based on the decay rate of hitting times of neighborhoods around zero.

Findings

If the expected hitting time of small neighborhoods does not grow faster than the inverse of their size, the Lévy transformation is strongly mixing.

The paper establishes a link between the decay rate of hitting times and ergodic properties.

Conditions are formulated in terms of a stationary sequence evaluated at time one.

Abstract

We propose a possible way of attacking the question posed originally by Daniel Revuz and Marc Yor in their book published in 1991. They were asking whether the L\'evy transformation of the Wiener--space is ergodic. Our main results are formulated in terms of a strongly stationary sequence of random variables obtained by evaluating the iterated paths at time one. Roughly speaking, this sequence has to approach zero "sufficiently fast". For example, one of our results states that if the expected hitting time of small neighbourhoods of the origin do not growth faster then the inverse of the size of these sets then the L\'evy transformation is strongly mixing, hence ergodic.