Fractal Dimensions of Rough Differential Equations Driven by Fractional Brownian Motions

Abstract

In this work we study fractal properties of rough differential equations driven by a fractional Brownian motions with Hurst parameter $H>\frac{1}{4}$. In particular, we show that the Hausdorff dimension of the sample paths of the solution is $\min\{d,\frac{1}{H}\}$ and that the Hausdorff dimension of the level set $L_x=\{ t\in[\epsilon,1]: X_t=x\}$ is $1-dH$ with positive probability when $d<\frac{1}{H}$