On the radius of self-repellent fractional Brownian motion

Abstract

We study the radius $R_T$ of a self-repellent fractional Brownian motion ${\left\{B_t^H\right\}}_{0\le t\le T}$ taking values in ${\mathbb{R}}^d$. Our sharpest result is for $d=1$, where we find that with high probability, \begin{equation*} R_T \asymp T^ν, \quad \text{with $\nu =\frac{2}{3}\left(1+H\right)$.} \end{equation*} For $d>1$, we provide upper and lower bounds for the exponent $\nu$, but these bounds do not match.