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

100%

Peigné M., Woess W.

Colloquium Mathematicum

2011

tom 125

nr 1

55-81

In this continuation of the preceding paper (Part I), we consider a sequence $(Fₙ)_{n≥0}$ of i.i.d. random Lipschitz mappings 𝖷 → 𝖷, where 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) $Xₙ^{x} = Fₙ ∘ ... ∘ F₁(x)$ starting at x ∈ 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 𝖷 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⁺⁎.

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

100%

Peigné M., Woess W.

Colloquium Mathematicum

2011

tom 125

nr 1

31-54

Consider a proper metric space 𝖷 and a sequence $(Fₙ)_{n≥0}$ of i.i.d. random continuous mappings 𝖷 → 𝖷. It induces the stochastic dynamical system (SDS) $Xₙ^{x} = Fₙ ∘ ... ∘ F₁(x)$ starting at x ∈ 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 In the present first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic iterations.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 Fₙ are contractions and, in particular, discuss recurrence criteria and their sharpness for the reflected random walk.

Ograniczanie wyników

2 Colloquium Mathematicum

2 Peigné M.

2 Woess W.

2 2011

Wyniki wyszukiwania

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

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