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 geometric fractional Brownian path $\boldsymbol{\omega}$ with Hurst index $H\in(\frac{1}{3},\frac{1}{2}]$. We first construct approximation $\boldsymbol{\omega}_{\delta}$ probabilistic arguments, then using theory obtain solution on any finite interval. Finally, both original system approximative generate continuous random dynamical systems $\varphi$ $\varphi^{\delta}$. As consequence solution, $\varphi^{\delta}$ converges as $\delta\rightarrow 0$.