Distortion inequality for a Markov operator generated by a randomly perturbed family of Markov Maps in ℝ d

Author:

Bugiel Peter1,Wędrychowicz Stanisław2,Rzepka Beata2

Affiliation:

1. Faculty of Mathematics and Computer Science, Jagiellonian University Cracow ( Kraków ), Poland

2. Department of Nonlinear Analysis, Rzeszów University of Technology, al . Powstańców Warszawy 8, 35-959 Rzeszów , Poland

Abstract

Abstract Asymptotic properties of the sequences (a) { P j } j = 1 $\{P^{j}\}_{j=1}^{\infty}$ and (b) { j 1 i = 0 j 1 P i } j = 1 $\{ j^{-1} \sum _{i=0}^{j-1} P^{i}\}_{j=1}^{\infty}$ are studied for gG = {fL 1(I) : f ≥ 0 and ‖f ‖ = 1}, where P : L 1(I) → L 1(I) is a Markov operator defined by P f := P y f d p ( y ) $Pf:= \int P_{y}f\, dp(y) $ for fL 1; {Py } y∈Y is the family of the Frobenius-Perron operators associated with a family {φy } y∈Y of nonsingular Markov maps defined on a subset I ⊆ ℝ d ; and the index y runs over a probability space (Y, Σ(Y), p). Asymptotic properties of the sequences (a) and (b), of the Markov operator P, are closely connected with the asymptotic properties of the sequence of random vectors x j = φ ξ j ( x j 1 ) $x_{j}=\varphi_{\xi_{j}}(x_{j-1})$ for j = 1,2, . . .,where { ξ j } j = 1 $\{\xi_{j}\}_{j=1}^{\infty}$ is a sequence of Y-valued independent random elements with common probability distribution p. An operator-theoretic analogue of Rényi’s Condition is introduced for the family {Py } y∈Y of the Frobenius-Perron operators. It is proved that under some additional assumptions this condition implies the L 1- convergence of the sequences (a) and (b) to a unique g 0G. The general result is applied to some families {φy } y∈Y of smooth Markov maps in ℝ d .

Publisher

Walter de Gruyter GmbH

Subject

Analysis

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