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 → Ω is an automorphism of the probability space Ω and f(ω, ·) 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(ω) of a random closed cone K(ω) in a finite-dimensional linear space into the cone K(Tω). Under the assumptions of monotonicity and homogeneity of D(ω), we prove the existence of scalar and vector measurable functions α(ω) > 0 and x(ω) ∈ K(ω) satisfying the equation α(ω)x(Tω) = D(ω)x(ω) almost surely.