# Stochastic nonlinear Perron–Frobenius theorem

Stochastic nonlinear Perron–Frobenius theorem We establish a stochastic nonlinear analogue of the Perron–Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The result is formulated in terms of an automorphism T of a probability space and a random transformation D of the non-negative cone of an n-dimensional Euclidean space. Under assumptions of monotonicity and homogeneity of D, we prove the existence of scalar and vector measurable functions α > 0 and x > 0 satisfying the equation αTx = D(x) almost surely. We apply the result obtained to the analysis of a class of random dynamical systems arising in mathematical economics and finance (von Neumann–Gale dynamical systems). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Stochastic nonlinear Perron–Frobenius theorem

, Volume 14 (1) – Jan 8, 2009
15 pages

/lp/springer_journal/stochastic-nonlinear-perron-frobenius-theorem-0rQyZH4Yaw
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-008-0003-2
Publisher site
See Article on Publisher Site

### Abstract

We establish a stochastic nonlinear analogue of the Perron–Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The result is formulated in terms of an automorphism T of a probability space and a random transformation D of the non-negative cone of an n-dimensional Euclidean space. Under assumptions of monotonicity and homogeneity of D, we prove the existence of scalar and vector measurable functions α > 0 and x > 0 satisfying the equation αTx = D(x) almost surely. We apply the result obtained to the analysis of a class of random dynamical systems arising in mathematical economics and finance (von Neumann–Gale dynamical systems).

### Journal

PositivitySpringer Journals

Published: Jan 8, 2009

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