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Rough Path Theory to approximate Random Dynamical Systems

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 $ω$ 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 $ϕ_0$. 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 $ω$. The latter differential equation generates a random dynamical system $ϕ_δ$ as well. We show the convergence of the random dynamical system $ϕ_δ$ to $ϕ_0$ for $δ\to 0$ in Hölder norm.