# Principal Floquet subspaces and exponential separations of type II with applications to random delay differential equations

## TL;DR

This paper introduces a new type of exponential separation, called type II, for positive random linear dynamical systems, with applications to scalar linear random delay differential equations with finite delay.

## Contribution

It develops the concept of exponential separation of type II and demonstrates its existence under weakened assumptions, applying it to random delay differential equations.

## Key findings

- Existence of exponential separation of type II established.
- Application to scalar linear random delay differential equations.
- Provides a framework for analyzing nonautonomous random systems with delay.

## Abstract

This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to nonautonomous random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.