Stochastic dynamical systems with weak contractivity properties (with a chapter featuring results of Martin Benda)

TL;DR

This paper investigates the existence, uniqueness, recurrence, and ergodicity of invariant measures in stochastic dynamical systems driven by i.i.d. random continuous mappings, extending Benda's work on local contractivity and applying results to reflected affine recursions.

Contribution

It improves and completes Benda's unpublished work on local contractivity and develops new criteria for recurrence and ergodicity in Lipschitz stochastic systems.

Findings

Established conditions for invariant measure existence and uniqueness.

Provided sharp recurrence criteria for reflected random walks.

Applied results to reflected affine stochastic recursions on the non-negative half-line.

Abstract

Consider a proper metric space X and a sequence of i.i.d. random continuous mappings F_n from X to X. It induces the stochastic dynamical system (SDS) X_n^x = F_n(X_{n-1}^x) starting at x in X. In this paper, we study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic iterations.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 F_n are contractions and, in particular, discuss recurrence criteria and their sharpness for reflected random walk. In the second part, we consider the case where the F_n are Lipschitz mappings. The main results concern the case when the associated Lipschitz constants are log-centered. Prinicpal tools are the Chacon-Ornstein theorem and a hyperbolic extension of the space X as well as the process (X_n^x). The results are applied to the reflected affine stochastic recursion on the non-negative half-line.