On unbounded invariant measures of stochastic dynamical systems

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.