The Annals of Probability

On unbounded invariant measures of stochastic dynamical systems

Sara Brofferio and Dariusz Buraczewski

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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.

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Ann. Probab. Volume 43, Number 3 (2015), 1456-1492.

First available in Project Euclid: 5 May 2015

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Zentralblatt MATH identifier

Primary: 37Hxx: Random dynamical systems [See also 15B52, 34D08, 34F05, 47B80, 70L05, 82C05, 93Exx] 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60K05: Renewal theory 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Stochastic recurrence equation stochastic dynamical system invariant measure affine recursion reflected random walk Poisson equation


Brofferio, Sara; Buraczewski, Dariusz. On unbounded invariant measures of stochastic dynamical systems. Ann. Probab. 43 (2015), no. 3, 1456--1492. doi:10.1214/13-AOP903.

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