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

Marc Peigné, and Wolfgang Woess. "Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings." Colloquium Mathematicae 125.1 (2011): 55-81. <http://eudml.org/doc/286398>.

@article{MarcPeigné2011,
abstract = {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⁺⁎.},
author = {Marc Peigné, Wolfgang Woess},
journal = {Colloquium Mathematicae},
keywords = {stochastic iterated function system; local contractivity; recurrence; invariant measure; ergodicity; random Lipschitz mappings; reflected affine stochastic recursion},
language = {eng},
number = {1},
pages = {55-81},
title = {Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings},
url = {http://eudml.org/doc/286398},
volume = {125},
year = {2011},
}

TY - JOUR
AU - Marc Peigné
AU - Wolfgang Woess
TI - Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings
JO - Colloquium Mathematicae
PY - 2011
VL - 125
IS - 1
SP - 55
EP - 81
AB - 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⁺⁎.
LA - eng
KW - stochastic iterated function system; local contractivity; recurrence; invariant measure; ergodicity; random Lipschitz mappings; reflected affine stochastic recursion
UR - http://eudml.org/doc/286398
ER -