Stochastic fixed points and nonlinear Perron--Frobenius theorem
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 estab...
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| Veröffentlicht in: | Proceedings of the American Mathematical Society 2018-10, Vol.146 (10), p.4315-4330 |
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| container_title | Proceedings of the American Mathematical Society |
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| creator | BABAEI, E. EVSTIGNEEV, I. V. PIROGOV, S. A. |
| description | 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. |
| doi_str_mv | 10.1090/proc/14075 |
| format | Article |
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A.</creator><general>American Mathematical Society</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20181001</creationdate><title>Stochastic fixed points and nonlinear Perron--Frobenius theorem</title><author>BABAEI, E. ; EVSTIGNEEV, I. V. ; PIROGOV, S. A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a319t-205b719e3723a6b2b7d5ac627b0dd14f8863f1fc415d42f790ff944ea8de90873</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>B. ANALYSIS</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>BABAEI, E.</creatorcontrib><creatorcontrib>EVSTIGNEEV, I. V.</creatorcontrib><creatorcontrib>PIROGOV, S. 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| source | American Mathematical Society Publications (Freely Accessible); JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; American Mathematical Society Publications; EZB-FREE-00999 freely available EZB journals |
| subjects | B. ANALYSIS |
| title | Stochastic fixed points and nonlinear Perron--Frobenius theorem |
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