Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings

Marc Peigné¹ ; Wolfgang Woess²

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

Colloquium Mathematicum, Tome 125 (2011) no. 1, pp. 55-81.

Résumé

In this continuation of the preceding paper (Part I), we consider a sequence $(F_n)_{n\ge 0}$ of i.i.d. random Lipschitz mappings $\mathsf X \to \mathsf X$, where $\mathsf X$ is a proper metric space. We investigate existence and uniqueness of invariant measures, as well as recurrence and ergodicity of the induced stochastic dynamical system (SDS) $X_n^x = F_n \circ \dots \circ F_1(x)$ starting at $x \in \mathsf X$. The main results concern the case when the associated Lipschitz constants are log-centered. Principal tools are local contractivity, as considered in detail in Part I, the Chacon–Ornstein theorem and a hyperbolic extension of the space $\mathsf X$ as well as the process $(X_n^x)$.The results are applied to a class of examples, namely, the reflected affine stochastic recursion given by $X_0^x=x \ge 0$ and $X_n^x = |A_nX_{n-1}^x - B_n|$, where $(A_n,B_n)$ is a sequence of two-dimensional i.i.d. random variables with values in $\mathbb R^+_* \times \mathbb R^+_*$.

Détail

DOI : 10.4064/cm125-1-5

Keywords: continuation preceding paper part nbsp consider sequence random lipschitz mappings mathsf mathsf where mathsf proper metric space investigate existence uniqueness invariant measures recurrence ergodicity induced stochastic dynamical system sds circ dots circ starting mathsf main results concern associated lipschitz constants log centered principal tools local contractivity considered detail part nbsp chacon ornstein theorem hyperbolic extension space mathsf process results applied class examples namely reflected affine stochastic recursion given n where n sequence two dimensional random variables values mathbb * times mathbb *