Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity

Volume 125 / 2011

Marc Peigné, Wolfgang Woess Colloquium Mathematicum 125 (2011), 31-54 MSC: 37Hxx, 60G50, 60J05. DOI: 10.4064/cm125-1-4


Consider a proper metric space $\mathsf{X}$ and a sequence $(F_n)_{n\ge 0}$ of i.i.d. random continuous mappings $\mathsf{X} \to \mathsf{X}$. It induces the stochastic dynamical system (SDS) $X_n^x = F_n \circ \dots \circ F_1(x)$ starting at $x \in \mathsf{X}$. In this and the subsequent paper, we study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process.

In the present first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic iterations. We consider the case when the $F_n$ are contractions and, in particular, discuss recurrence criteria and their sharpness for the reflected random walk.


  • Marc PeignéLaboratoire de Mathématiques
    et Physique Théorique
    Université François Rabelais Tours
    Fédération Denis Poisson – CNRS
    Parc de Grandmont
    37200 Tours, France
  • Wolfgang WoessInstitut für Mathematische Strukturtheorie
    (Math C)
    Technische Universität Graz
    Steyrergasse 30, A-8010 Graz, Austria

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