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
MathSciNet review:
3834661
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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
MR Author ID:
1040683
Email:
esmaeil.babaeikhezerloo@postgrad.manchester.ac.uk
I. V. Evstigneev
Affiliation:
Department of Economics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
MR Author ID:
210292
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
MR Author ID:
231708
Email:
s.a.pirogov@bk.ru
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