On the radius of self-repellent fractional Brownian motion

TL;DR

This paper investigates the growth rate of the radius of self-repellent fractional Brownian motion in various dimensions, establishing precise asymptotics in one dimension and bounds in higher dimensions.

Contribution

It provides a sharp asymptotic result for the radius in one dimension and bounds for higher dimensions, advancing understanding of self-repellent fractional Brownian motion.

Findings

In 1D, radius scales as T^{(2/3)(1+H)} with high probability.

For higher dimensions, bounds on the radius exponent are established.

The bounds in higher dimensions do not match, indicating areas for further research.

Abstract

We study the radius $R_T$ 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{equation*} R_T \asymp T^\nu, \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.