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

Marc Peigné¹ ; Wolfgang Woess²

¹Laboratoire de Mathématiques et Physique Théorique Université François Rabelais Tours Fédération Denis Poisson – CNRS Parc de Grandmont 37200 Tours, France
²Institut für Mathematische Strukturtheorie (Math C) Technische Universität Graz Steyrergasse 30, A-8010 Graz, Austria

Colloquium Mathematicum, Tome 125 (2011) no. 1, pp. 31-54

Résumé

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as well as recurrence and ergodicity of this process.In the present first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic iterations.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.

Détail

DOI : 10.4064/cm125-1-4

Keywords: consider proper metric space mathsf sequence random continuous mappings mathsf mathsf induces stochastic dynamical system sds circ dots circ starting mathsf subsequent paper study existence uniqueness invariant measures recurrence ergodicity process present first part elaborate improve complete unpublished work martin benda local contractivity which merits publicity provides important tool studying stochastic iterations consider contractions particular discuss recurrence criteria their sharpness reflected random walk