Invariance and Monotonicity for Stochastic Delay Differential Equations

TL;DR

This paper investigates invariance and monotonicity properties of stochastic delay differential equations, providing conditions for invariance, a comparison principle, and demonstrating that such systems can generate order-preserving random dynamical systems.

Contribution

It offers new sufficient conditions for invariance and monotonicity in stochastic delay systems, extending the understanding of their dynamical behavior.

Findings

Provided conditions for invariance of closed subsets in ^d

Established a comparison principle for stochastic delay equations

Showed these systems can generate monotone random dynamical systems

Abstract

We study invariance and monotonicity properties of Kunita-type stochastic differential equations in $\RR^d$ with delay. Our first result provides sufficient conditions for the invariance of closed subsets of $\RR^d$. Then we present a comparison principle and show that under appropriate conditions the stochastic delay system considered generates a monotone (order-preserving) random dynamical system. Several applications are considered.