The Multiple Points of Fractional Brownian Motion

2020/03/06 by Mark Landry, Cheuk Yin Lee, Landry, Mark +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60G22 #Chaos control and synchronization #Complex Systems and Time Series Analysis #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2003.03005

openalex publication_date 2020/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 ℝ^d with Hurst index H such that Hd=1, and E is a fixed, nonempty compact set in ℝ^d with positive capacity with respect to the function ϕ(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 ϕ(s) = s^-k(d-1/H).

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