Synchronization of stochastic dissipative differential equation driven by fractional Brownian motions

Abstract: In this paper, we study a class of dissipative stochastic differential equations driven by nonlinear multiplicative fractional Brownian noise with Hurst index $H \in \left(\frac{1}{3},\frac{1}{2})\cup(\frac{1}{2}, 1\right) $. We establish the well-posedness of the associated coupled stochastic differential equations and prove synchronization in the sense of trajectories. Our approach relies on the Doss-Sussmann transformation, which enables us to extend existing results for additive and linear noise to the case of nonlinear multiplicative fractional Brownian noise. The findings provide new insights into the synchronization of dissipative systems under fractional noise perturbations.