Synchronization of Differential Equations Driven by Linear Multiplicative Fractional Brownian Motion

Abstract

This paper is devoted to the synchronization of stochastic differential equations driven by the linear multiplicative fractional Brownian motion with Hurst parameter $H\in (\frac{1}{2},1)$. We firstly prove that the equation has a unique stationary solution which generates a random dynamical system. Moreover the system has the pathwise singleton sets random attractor. Next we show up the synchronization of solutions of two coupled differential equations. At the end, we discuss two specific situations and provide the corresponding synchronization results.