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Stochastic fixed points and nonlinear Perron-Frobenius theorem


Authors: E. Babaei, I. V. Evstigneev and S. A. Pirogov
Journal: Proc. Amer. Math. Soc. 146 (2018), 4315-4330
MSC (2010): Primary 37H10, 37H15; Secondary 37H05, 37H99
DOI: https://doi.org/10.1090/proc/14075
Published electronically: June 13, 2018
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Abstract: We provide conditions for the existence of measurable solutions to the equation $ \xi (T\omega )=f(\omega ,\xi (\omega ))$, where $ T:\Omega \rightarrow \Omega $ is an automorphism of the probability space $ \Omega $ and $ f(\omega ,\cdot )$ is a strictly nonexpansive mapping. We use results of this kind to establish a stochastic nonlinear analogue of the Perron-Frobenius theorem on eigenvalues and eigenvectors of a positive matrix. We consider a random mapping $ D(\omega )$ of a random closed cone $ K(\omega )$ in a finite-dimensional linear space into the cone $ K(T\omega )$. Under the assumptions of monotonicity and homogeneity of $ D(\omega )$, we prove the existence of scalar and vector measurable functions $ \alpha (\omega )>0$ and $ x(\omega )\in K(\omega )$ satisfying the equation $ \alpha (\omega )x(T\omega )=D(\omega )x(\omega )$ almost surely.


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Additional Information

E. Babaei
Affiliation: Department of Economics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
Email: esmaeil.babaeikhezerloo@postgrad.manchester.ac.uk

I. V. Evstigneev
Affiliation: Department of Economics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
Email: igor.evstigneev@manchester.ac.uk

S. A. Pirogov
Affiliation: Department of Mechanics and Mathematics, Moscow State University, ul. Lenĭnskiye Gory, 1, Moscow, Russia, 119234 –and– the Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetny 19-1, Moscow, 127051, Russia
Email: s.a.pirogov@bk.ru

DOI: https://doi.org/10.1090/proc/14075
Keywords: Random dynamical systems, contraction mappings, Perron--Frobenius theory, nonlinear cocycles, stochastic equations, random monotone mappings, Hilbert--Birkhoff metric
Received by editor(s): January 25, 2017
Received by editor(s) in revised form: December 29, 2017
Published electronically: June 13, 2018
Communicated by: Yingfei Yi
Article copyright: © Copyright 2018 American Mathematical Society

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