On the Radius of Self-Repellent Fractional Brownian Motion

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