On unbounded invariant measures of stochastic dynamical systems

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 $A_{n}x$, for some random positive number $A_{n}$ [the main example is the affine stochastic recursion $\Psi_{n}(x)=A_{n}x+B_{n}$]. 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.