Synchronization by noise for stochastic differential equations driven by fractional Brownian motion

TL;DR

This paper studies how noise can induce synchronization in stochastic differential equations driven by fractional Brownian motion, especially when the system exhibits negative top Lyapunov exponents, using advanced stochastic dynamical systems tools.

Contribution

It provides a novel analysis of synchronization by noise in non-Markovian SDEs driven by fractional Brownian motion, including support characterization of invariant measures.

Findings

Weak synchronization occurs when the top Lyapunov exponent is negative.

Support of invariant measures is characterized in a non-Markovian setting.

Tools from stochastic and random dynamical systems are employed.

Abstract

We investigate synchronization by noise for stochastic differential equations (SDEs) driven by a fractional Brownian motion (fbm) with Hurst index $H\in(0,1)$. Provided that the SDE has a negative top Lyapunov exponent, we show that a weak form of synchronization occurs. To this aim we use tools from stochastic dynamical systems, random dynamical systems and a support theorem for SDEs driven by fractional noise.~In particular, we characterize the support of an invariant measure of a random dynamical system in a non-Markovian setting.