Stochastic Fixed Points and Nonlinear Perron-Frobenius Theorem

Abstract

We provide conditions for the existence of measurable solutions to the equation $\xi (T\omega )=f(\omega ,\xi (\omega ))$, where $T:\Omega \to \Omega$ 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.