Stochastic Fixed Points and Nonlinear Perron-Frobenius Theorem

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

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