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

Qiyong Cao; Hongjun Gao; Bjorn Schmalfuss

doi:10.1016/j.jde.2023.02.031

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arXiv · 2210.04239

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

Qiyong Cao·Hongjun Gao·Bjorn Schmalfuss

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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 $\delta\rightarrow 0$.

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BibTeX RIS

Qiyong Cao, Hongjun Gao, Bjorn Schmalfuss. 2022-10-09. Wong-Zakai type approximations of rough random dynamical systems by smooth noise. https://doi.org/10.1016/j.jde.2023.02.031

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