Rough Path Theory to approximate Random Dynamical Systems

Abstract

We consider the rough differential equation $dY=f(Y)d\bm \om$ where $\bm \om=(ω,\bbomega)$ is a rough path defined by a Brownian motion $\omega$ on $\RR^m$. Under the usual regularity assumption on $f$, namely $f\in C^3_b (\RR^d, \RR^{d\times m})$, the rough differential equation has a unique solution that defines a random dynamical system ${\varphi }_0$. On the other hand, we also consider an ordinary random differential equation $dY_δ=f(Y_δ)dω_\de$, where $ω_\de$ is a random process with stationary increments and continuously differentiable paths that approximates $\omega$. The latter differential equation generates a random dynamical system ${\varphi }_{\delta }$ as well. We show the convergence of the random dynamical system ${\varphi }_{\delta }$ to ${\varphi }_0$ for $\delta \to 0$ in Hölder norm.

23 pages