## 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 language | English |
---|---|

Pages (from-to) | 4315-4330 |

Number of pages | 16 |

Journal | Proceedings of the American Mathematical Society |

Volume | 146 |

Early online date | 13 Jun 2018 |

DOIs | |

Publication status | Published - 2018 |

## Keywords

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