Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37H10, 37H15, 37H05, 37H99

Retrieve articles in all journals with MSC (2010): 37H10, 37H15, 37H05, 37H99


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