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

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 characterize the support of an invariant measure of a random dynamical system in a non-Markovian setting.