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

Abstract

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 $\mathbf{\omega }$ with Hurst index $H\in (\frac{1}{3},\frac{1}{2}]$. We first construct the approximation ${\mathbf{\omega }}_{\delta }$ of $\mathbf{\omega }$ 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 $\phi$ and ${\phi }^{\delta }$. As a consequence of the Wong-Zakai approximation of the solution, ${\phi }^{\delta }$ converges to $\phi$ as $\delta \to 0$.