On the iterations of some random functions with Lipschitz number one

TL;DR

This paper analyzes the dynamics of certain Lipschitz continuous random functions through Markov chains, establishing convergence properties and fixed points, with implications for understanding their long-term behavior.

Contribution

It introduces a Markov chain representation for the iterations of Lipschitz-one functions and characterizes convergence and fixed points in the space of probability measures.

Findings

Proves convergence of the Markov chain under mild conditions.

Shows Wasserstein metric does not increase under the induced iterations.

Identifies fixed points of an associated nonlinear operator.

Abstract

For the iterations of $x\mapsto |x-\theta|$ random functions with Lipschitz number one, we represent the dynamics as a Markov chain and prove its convergence under mild conditions. We also demonstrate that the Wasserstein metric of any two measures will not increase after the corresponding induced iterations for measures and identify conditions under which a polynomial convergence rate can be achieved in this metric. We also consider an associated nonlinear operator on the space of probability measures and identify its fixed points through an detailed analysis of their characteristic functions.