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

Marc Peigné ; Wolfgang Woess

Colloquium Mathematicae, Tome 122 (2011), p. 55-81 / Harvested from The Polish Digital Mathematics Library

Résumé

In this continuation of the preceding paper (Part I), we consider a sequence $(Fₙ)_{n≥0}$ of i.i.d. random Lipschitz mappings → , where 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) $Xₙ^{x} = Fₙ ∘ ... ∘ F₁(x)$ starting at 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 as well as the process $(Xₙ^{x})$. The results are applied to a class of examples, namely, the reflected affine stochastic recursion given by $X₀^{x} = x ≥ 0$ and $Xₙ^{x} = |AₙX_{n-1}^{x} - Bₙ|$, where (Aₙ,Bₙ) is a sequence of two-dimensional i.i.d. random variables with values in ℝ⁺⁎ × ℝ⁺⁎.

Détail

Publié le : 2011-01-01

Zbl 1260.37026

EUDML-ID : urn:eudml:doc:286398