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

TL;DR

This paper develops smooth Wong-Zakai approximations for rough differential equations driven by fractional Brownian rough paths, demonstrating convergence of the approximated solutions to the original system.

Contribution

It introduces a probabilistic construction of smooth approximations for fractional Brownian rough paths and proves the convergence of the associated random dynamical systems.

Findings

Wong-Zakai approximations converge as the smoothing parameter tends to zero.

Constructed smooth approximations are stationary and preserve key properties.

Established convergence results for solutions and dynamical systems.

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