STOCHASTIC FIXED POINTS AND NONLINEAR PERRON-FROBENIUS THEOREM

Esmaeil Babaei Khezerloo, Igor Evstigneev, S. A. Pirogov

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Abstract

We provide conditions for the existence of measurable solutions to the equation ξ(Tω) = f(ω, ξ(ω)), where T: Ω → Ω is an automorphism of the probability space Ω and f(ω, ·) 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(ω) of a random closed cone K(ω) in a finite-dimensional linear space into the cone K(Tω). Under the assumptions of monotonicity and homogeneity of D(ω), we prove the existence of scalar and vector measurable functions α(ω) > 0 and x(ω) ∈ K(ω) satisfying the equation α(ω)x(Tω) = D(ω)x(ω) almost surely.

Original languageEnglish
Pages (from-to)4315-4330
Number of pages16
JournalProceedings of the American Mathematical Society
Volume146
Early online date13 Jun 2018
DOIs
Publication statusPublished - 2018

Keywords

  • Random dynamical systems
  • Contraction mappings
  • Perron-Frobenius theory
  • nonlinear cocycles
  • Stochastic equations
  • andom monotone mappings
  • Hilbert-Birkho metric

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