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Optimal transportation and stationary measures for iterated function systems

Part of: Markov processes Classical measure theory

Published online by Cambridge University Press:  28 June 2021

BENOÎT R. KLOECKNER
BENOÎT R. KLOECKNER*
Affiliation:
Univ Paris Est Creteil, CNRS, LAMA, F-94010 Creteil, France Univ Gustave Eiffel, LAMA, F-77447 Marne-la-Vallée, France. e-mail: benoit.kloeckner@u-pec.fr
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Abstract

In this paper we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measures of Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove generalised moment bounds from which tail estimates can be deduced, consider the convergence of the empirical measure of an associated Markov chain, and prove in many cases the Lipschitz continuity of the stationary measure when the system is perturbed, with as a consequence a “linear response formula” at almost every parameter of the perturbation.

MSC classification

Primary: 28A80: Fractals
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 1 , July 2022 , pp. 163 - 187
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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