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 ω with Hurst index H∈(13,12]. We first construct the approximation ωδ of ω by probabilistic arguments, and then use the rough path theory to obtain the Wong-Zakai approximation for the solution on any finite interval. Finally, both the original system and the approximative system generate a continuous random dynamical systems φ and φδ. As a consequence of the Wong-Zakai approximation of the solution, φδ converges to φ as δ→0.