Invariance and monotonicity  for stochastic delay differential equations

Igor Chueshov; Michael Scheutzow

doi:10.3934/dcdsb.2013.18.1533

Article Contents

2013, Volume 18, Issue 6: 1533-1554. Doi: 10.3934/dcdsb.2013.18.1533

This issue Previous Article A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations Next Article Exponential stability for a class of linear hyperbolic equations with hereditary memory

Invariance and monotonicity for stochastic delay differential equations

Igor Chueshov^1, and
Michael Scheutzow^2,

1.
Department of Mechanics and Mathematics, Kharkov National University, Kharkov, 61022
2.
Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin

Received: January 2012

Revised: March 2012

Published: August 2013

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Abstract

We study invariance and monotonicity properties of Kunita-type sto-chastic differential equations in $\mathbb{R}^d$ with delay. Our first result provides sufficient conditions for the invariance of closed subsets of $\mathbb{R}^d$. 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.

Keywords:

Stochastic delay/functional differential equation,
stochastic flow,
random dynamical system,
invariance,
monotonicity,
random attractor.

Mathematics Subject Classification: Primary: 34K50, 37H10; Secondary: 37H10.

Citation:

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