Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory

TL;DR

This paper develops a general theoretical framework for principal Lyapunov exponents and Floquet subspaces in positive random dynamical systems within ordered Banach spaces, extending classical concepts to stochastic settings.

Contribution

It introduces generalized notions of principal Floquet subspaces, Lyapunov exponents, and exponential separations for positive random dynamical systems, broadening classical deterministic theories.

Findings

Existence of generalized principal Floquet subspaces under broad conditions

Definition of a generalized principal Lyapunov exponent for random systems

Establishment of generalized exponential separation in the stochastic setting

Abstract

This is the first of a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. It focuses on the development of general theory. First, the notions of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations for general positive random dynamical systems in ordered Banach spaces are introduced, which extend the classical notions of principal Floquet subspaces, principal Lyapunov exponents, and exponential separations for strongly positive deterministic systems in strongly ordered Banach to general positive random dynamical systems in ordered Banach spaces. Under some quite general assumptions, it is then shown that a positive random dynamical system in an ordered Banach space admits a family of generalized principal Floquet subspaces, a generalized principal Lyapunov exponent, and a generalized exponential separation. We will consider in the forthcoming parts applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations, including random Leslie matrix models, random cooperative systems of ordinary differential equations, and random parabolic equations.