Stochastic fixed points and nonlinear Perron–Frobenius theorem

Abstract

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