The Multiple Points of Fractional Brownian Motion

Nils Tongring (1987) proved sufficient conditions for a compact set to contain $k$-tuple points of a Brownian motion. In this paper, we extend these findings to the fractional Brownian motion. Using the property of strong local nondeterminism, we show that if $B$ is a fractional Brownian motion in $\mathbb{R}^d$ with Hurst index $H$ such that $Hd=1$, and $E$ is a fixed, nonempty compact set in $\mathbb{R}^d$ with positive capacity with respect to the function $\phi(s) = (\log_+(1/s))^k$, then $E$ contains $k$-tuple points with positive probability. For the $Hd > 1$ case, the same result holds with the function replaced by $\phi(s) = s^{-k(d-1/H)}$.