Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings
Volume 125 / 2011
Abstract
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 $\mathbb R^+_* \times \mathbb R^+_*$.
