Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. III. Parabolic equations and delay systems

TL;DR

This paper applies the general theory of principal Lyapunov exponents and Floquet spaces to positive random dynamical systems derived from parabolic equations and delay systems, establishing the existence of measurable principal subspaces and exponents.

Contribution

It extends the theory to infinite-dimensional systems from parabolic and delay equations, demonstrating the existence of measurable principal Floquet subspaces and Lyapunov exponents.

Findings

Existence of measurable principal Floquet subspaces for random parabolic equations.

Existence of generalized principal Lyapunov exponents in delay systems.

Application of theory to infinite-dimensional ordered Banach spaces.

Abstract

This is the third part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors' paper "Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory," Trans. Amer. Math. Soc. 365 (2013), pp. 5329-5365, to positive continuous-time random dynamical systems on infinite dimensional ordered Banach spaces arising from random parabolic equations and random delay systems. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by random parabolic equations and by cooperative systems of linear delay differential equations admit measurable families of generalized principal Floquet subspaces, and generalized principal Lyapunov exponents.