Stochastic Nonlinear Perron-Frobenius theorem

The stochastic nonlinear analogue of Perron-Frobenius theorem on eigenvectors and eigenvalues of positive matrices is considered. This result is about the pair, an automorphism T of probability space W and a random nonlinear map D(w) acting from X(w) to X(Tw), where X(w) is a random cone in a n-dimensional vector space. Under some assumptions of monotonicity and homogeneity of D(w) we prove the existence and uniqueness of a scalar function a(w)>0 and a vector function x(w) with values in X(w) such that a(w)x(Tw)=D(w,x(w)) a.e. Such results are applied in the analysis of random dynamical systems in physics, biology, economics, finances etc.
The talk is based on the paper E. Babaei, I.V. Evstigneev, S.A. Pirogov, Stochastic fixed points and nonlinear Perron-Frobenius theorem, Proceedings of the American Mathematical Society,2018 (in press), DOI: https://doi.org/10.1090/proc/14075.