Stochastic fixed points and nonlinear Perron–Frobenius theorem

Babaei, E.; Evstigneev, I.; Pirogov, S.

doi:10.1090/proc/14075

Stochastic fixed points and nonlinear Perron–Frobenius theorem
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by E. Babaei, I. V. Evstigneev and S. A. Pirogov PDF

Proc. Amer. Math. Soc. 146 (2018), 4315-4330 Request permission

Abstract:

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

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