On the Radius of Self-Repellent Fractional Brownian Motion

Abstract

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