On unbounded invariant measures of stochastic dynamical systems

We consider stochastic dynamical systems on R, that is, random processes defined by Xxn=.n(Xxn.1), Xx0=x, where .n are i.i.d. random continuous transformations of some unbounded closed subset of R. We assume here that .n behaves asymptotically like Anx, for some random positive number An [the main example is the affine stochastic recursion .n(x)=Anx+Bn]. Our aim is to describe invariant Radon measures of the process Xxn in the critical case, when ElogA1=0. We prove that those measures behave at infinity like dxx. 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.