Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-20T10:35:01.275Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ambrosio, L., Gigli, N. and Savaré, G.. Gradient flows: in metric spaces and in the space of probability measures. (Springer, 2008).10.1016/S1874-5717(07)80004-1CrossRefGoogle Scholar
Anckar, A. and Högnäs, G.. The fine structure of the stationary distribution for a simple Markov process. Probability on algebraic and geometric structures, Contemp. Math., vol. 668 (Amer. Math. Soc., Providence, RI, 2016), pp. 1–12.CrossRefGoogle Scholar
Barnsley, M. F., Demko, S. G., Elton, J. H. and Geronimo, J. S.. Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), no. 3, 367–394.Google Scholar
Barnsley, M. F., Devaney, R. L., Mandelbrot, B. B., Peitgen, H.-O., Saupe, D. and Voss, R. F.. The science of fractal images, (Springer-Verlag, New York, 1988), With contributions by Yuval Fisher and Michael McGuire.CrossRefGoogle Scholar
Barnsley, M. F. and Elton, J. H.. A new class of Markov processes for image encoding. Adv. in Appl. Probab. 20 (1988), no. 1, 14–32.CrossRefGoogle Scholar
Bergelson, V., Misiurewicz, M. and Senti, S.. Affine actions of a free semigroup on the real line. Ergodic Theory Dynam. Systems 26 (2006), no. 5, 1285–1305.CrossRefGoogle Scholar
Baladi, V. and Smania, D.. Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps. Ann. Sci. Éc. Norm. Sup. (4) 45 (2012), no. 6, 861–926 (2013).10.24033/asens.2179CrossRefGoogle Scholar
Barnsley, M. F. and Vince, A.. The chaos game on a general iterated function system. Ergodic Theory Dynam. Systems 31 (2011), no. 4, 1073–1079.CrossRefGoogle Scholar
Diaconis, P. and Freedman, D.. Iterated random functions. SIAM review 41 (1999), no. 1, 45–76.CrossRefGoogle Scholar
Dedecker, J. and Merlevède, F.. Behavior of the empirical Wasserstein distance in $${{\mathbb R}^d}$$ under moment conditions. Electron. J. Probab. 24 (2019).CrossRefGoogle Scholar
Elton, J. H.. An ergodic theorem for iterated maps. Ergodic Theory Dynam. Systems 7 (1987), no. 4, 481–488.10.1017/S0143385700004168CrossRefGoogle Scholar
Elton, J. H.. A multiplicative ergodic theorem for Lipschitz maps. Stochastic Process. Appl. 34 (1990), no. 1, 39–47.CrossRefGoogle Scholar
Fournier, N. and Guillin, A.. On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Related Fields 162 (2015), no. 3-4, 707–738.CrossRefGoogle Scholar
Forte, B. and Mendivil, F.. A classical ergodic property for IFS: a simple proof. Ergodic Theory Dynam. Systems 18 (1998), no. 3, 609–611.CrossRefGoogle Scholar
Fraser, J. M.. First and second moments for self-similar couplings and Wasserstein distances. Math. Nachr. 288 (2015), no. 17-18, 2028–2041.CrossRefGoogle Scholar
Giulietti, P., Kloeckner, B., Lopes, A. O. and Marcon, D.. The calculus of thermodynamical formalism. J. Eur. Math. Soc. (JEMS) 20 (2018), no. 10, 2357–2412.CrossRefGoogle Scholar
Galatolo, S., Monge, M. and Nisoli, I.. Rigorous approximation of stationary measures and convergence to equilibrium for iterated function systems. J. Phys. A 49 (2016), no. 27, 274001, 22.10.1088/1751-8113/49/27/274001CrossRefGoogle Scholar
Goldie, C. M.. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 (1991), no. 1, 126–166.CrossRefGoogle Scholar
Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), no. 5, 713–747.10.1512/iumj.1981.30.30055CrossRefGoogle Scholar
Iosifescu, M.. Iterated function systems. A critical survey, Math. Rep. (Bucur.) 11(61) (2009), no. 3, 181–229.Google Scholar
Joulin, A. and Ollivier, Y.. Curvature, concentration and error estimates for Markov chain Monte Carlo. Ann. Probab. 38 (2010), no. 6, 2418–2442.10.1214/10-AOP541CrossRefGoogle Scholar
Kesten, H.. Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973), no. 1, 207–248.CrossRefGoogle Scholar
Kevei, P.. A note on the Kesten-Grincevičius-Goldie theorem. Electron. Commun. Probab. 21 (2016), Paper No. 51, 12.CrossRefGoogle Scholar
Kloeckner, B. R.. Empirical measures: regularity is a counter-curse to dimensionality. arXiv:1802.04038, (2018).Google Scholar
Kloeckner, B. R.. Extensions with shrinking fibers. arXiv:1812.08437, (2018).Google Scholar
Madras, N. and Sezer, D.. Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances. Bernoulli 16 (2010), no. 3, 882–908.CrossRefGoogle Scholar
Nicol, M., Sidorov, N. and Broomhead, D.. On the fine structure of stationary measures in systems which contract-on-average. J. Theoret. Probab. 15 (2002), no. 3, 715–730.CrossRefGoogle Scholar
Ollivier, Y.. Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256 (2009), no. 3, 810–864.CrossRefGoogle Scholar
Peigné, M.. Iterated function systems and spectral decomposition of the associated Markov operator. Fascicule de probabilités, Publ. Inst. Rech. Math. Rennes, vol. 1993, Univ. Rennes I, Rennes, 1993, p. 28.Google Scholar
Pollicott, M.. Contraction in mean and transfer operators. Dyn. Syst. 16 (2001), no. 1, 97–106.CrossRefGoogle Scholar
Ruelle, D.. General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium. Phys. Lett. A 245 (1998), no. 3-4, 220–224.CrossRefGoogle Scholar
Ruelle, D.. A review of linear response theory for general differentiable dynamical systems. Nonlinearity 22 (2009), no. 4, 855–870.CrossRefGoogle Scholar
Solomyak, B.. On the random series $$\sum { \pm {\lambda ^n}} $$ (an Erdős problem). Ann. of Math. (2) 142 (1995), no. 3, 611–625.CrossRefGoogle Scholar
Silvestrov, D. S. and Stenflo, Ö.. Ergodic theorems for iterated function systems controlled by regenerative sequences. J. Theoret. Probab. 11 (1998), no. 3, 589–608.CrossRefGoogle Scholar
Steinsaltz, D.. Locally contractive iterated function systems. Ann. Probab. 27 (1999), no. 4, 1952–1979.CrossRefGoogle Scholar
Santos, S. I. and Walkden, C.. Distributional and local limit laws for a class of iterated maps that contract on average. Stoch. Dyn. 13 (2013), no. 2, 1250019, 28.CrossRefGoogle Scholar
Szarek, T.. Invariant measures for nonexpensive Markov operators on Polish spaces. Dissertationes Math. (Rozprawy Mat.) 415 (2003), 62, Dissertation, Polish Academy of Science, Warsaw, (2003).CrossRefGoogle Scholar
Varjú, P. P.. Recent progress on Bernoulli convolutions, European Congress of Mathematics. Eur. Math. Soc., Zürich, (2018), pp. 847–867.Google Scholar
Villani, C.. Optimal transport, Grund. Math. Wiss. [Fundamental Principles of Mathematical Sciences], vol. 338 (Springer-Verlag, Berlin, 2009), Old and new.Google Scholar
Walkden, C. P.. Invariance principles for iterated maps that contract on average. Trans. Amer. Math. Soc. 359 (2007), no. 3, 1081–1097.CrossRefGoogle Scholar