Stochastic Fixed Points and Nonlinear Perron-Frobenius Theorem

TL;DR

This paper extends the Perron-Frobenius theorem to stochastic nonlinear settings, establishing conditions for the existence of measurable eigenfunctions and eigenvalues for random nonlinear operators on cones.

Contribution

It introduces a stochastic nonlinear Perron-Frobenius theorem, providing conditions for measurable solutions to nonlinear eigenvalue problems in random environments.

Findings

Existence of measurable solutions to stochastic nonlinear eigenvalue equations.

Conditions under which random nonlinear operators have positive eigenfunctions.

Extension of Perron-Frobenius theory to stochastic, nonlinear, and cone-valued mappings.

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 non-expansive 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 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.