On unbounded invariant measures of stochastic dynamical systems

TL;DR

This paper investigates the behavior of invariant measures for stochastic dynamical systems on the real line, especially in the critical case where the expected logarithm of the multiplicative factor is zero, revealing measures that decay like 1/x at infinity.

Contribution

It extends previous results on affine recursions to a broader class of stochastic systems, including reflected walks and processes on [0,1], providing new insights into invariant measure behavior.

Findings

Invariant measures behave like dx/x at infinity.

Improved conditions for uniqueness of invariant measures.

Generalization to systems beyond affine recursions.

Abstract

We consider stochastic dynamical systems on ${\mathbb{R}}$, that is, random processes defined by $X_n^x=\Psi_n(X_{n-1}^x)$, $X_0^x=x$, where $\Psi _n$ are i.i.d. random continuous transformations of some unbounded closed subset of ${\mathbb{R}}$. We assume here that $\Psi_n$ behaves asymptotically like $A_nx$, for some random positive number $A_n$ [the main example is the affine stochastic recursion $\Psi_n(x)=A_nx+B_n$]. Our aim is to describe invariant Radon measures of the process $X_n^x$ in the critical case, when ${\mathbb{E}}\log A_1=0$. We prove that those measures behave at infinity like $\frac{dx}{x}$. We study also the problem of uniqueness of the invariant measure. We improve previous results known for the affine recursions and generalize them to a larger class of stochastic dynamical systems which include, for instance, reflected random walks, stochastic dynamical systems on the unit interval $[0,1]$, additive Markov processes and a variant of the Galton--Watson process.