## On unbounded invariant measures of stochastic dynamical systems.(English)Zbl 1352.37155

Summary: 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.

### MSC:

 37H10 Generation, random and stochastic difference and differential equations 60J05 Discrete-time Markov processes on general state spaces 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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