Stochastic fixed points and nonlinear Perron–Frobenius theorem

We provide conditions for the existence of measurable solutions to the equation xi(T omega) = f(omega, xi(omega)), where T : Omega -> Omega is an automorphism of the probability space Omega and f(omega, center dot) 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(omega) of a random closed cone K(omega) in a finite-dimensional linear space into the cone K(T.). Under the assumptions of monotonicity and homogeneity of D(Tau omega), we prove the existence of scalar and vector measurable functions alpha(omega) > 0 and x(omega) is an element of K(omega) satisfying the equation alpha(omega) x(Tau omega) = D(omega) x(omega) almost surely.