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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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