Wong-Zakai type approximations of rough random dynamical systems by smooth noise

This paper is devoted to the smooth and stationary Wong-Zakai approximations for a class of rough differential equations driven by a geometric fractional Brownian rough path $\boldsymbolω$ with Hurst index $H\in(\frac{1}{3},\frac{1}{2}]$. We first construct the approximation $\boldsymbolω_δ$ of $\boldsymbolω$ by probabilistic arguments, and then using the rough path theory to obtain the Wong-Zakai approximation for the solution on any finite interval. Finally, both the original system and approximative system generate a continuous random dynamical systems $\varphi$ and $\varphi^δ$. As a consequence of the Wong-Zakai approximation of the solution, $\varphi^δ$ converges to $\varphi$ as $δ\rightarrow 0$.