On random linear dynamical systems in a Banach space. I. Multiplicative Ergodic Theorem and Krein-Rutmann type Theorems

TL;DR

This paper develops Krein-Rutman type theorems for linear random dynamical systems in Banach spaces, linking co-invariant cones, dominated splittings, and Lyapunov exponents, extending classical results to infinite-dimensional settings.

Contribution

It introduces new Krein-Rutman type theorems for Banach space systems and establishes their relation to the Multiplicative Ergodic Theorem and dominated splittings.

Findings

Established (quasi)-equivalence between co-invariant cone families and dominated splittings.

Proved the equivalence relation in specific cases, such as rank-1 cones and certain Lyapunov exponent conditions.

Analyzed the interplay between Lyapunov exponents, cones, and splittings in the context of random dynamical systems.

Abstract

For linear random dynamical systems in a separable Banach space $X$, we derived a series of Krein-Rutman type Theorems with respect to co-invariant cone family with rank-$k$, which present a (quasi)-equivalence relation between the measurably co-invariant cone family and the measurably dominated splitting of $X$. Moreover, such (quasi)-equivalence relation turns out to be an equivalence relation whenever (i) $k=1$; or (ii) in the frame of the Multiplicative Ergodic Theorem with certain Lyapunov exponent being greater than the negative infinity. For the second case, we thoroughly investigated the relations between the Lyapunov exponents, the co-invariant cone family and the measurably dominated splitting for linear random dynamical systems in $X$.