Fractal Dimensions of Rough Differential Equations Driven by Fractional Brownian Motions

TL;DR

This paper investigates the fractal properties of solutions to rough differential equations driven by fractional Brownian motions, establishing formulas for the Hausdorff dimensions of sample paths and level sets based on the Hurst parameter.

Contribution

It provides new results on the Hausdorff dimensions of solution paths and level sets for rough differential equations driven by fractional Brownian motion, extending understanding of their fractal geometry.

Findings

Hausdorff dimension of solution paths is min{d, 1/H}

Hausdorff dimension of level sets is 1 - dH with positive probability when d < 1/H

Results apply for Hurst parameter H > 1/4

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 $d<\frac{1}{H}$