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...

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Bibliographic Details
Published in:arXiv.org 2020-03
Main Authors: Landry, Mark, Cheuk Yin Lee, Pearcy, Paige
Format: Article
Language:English
Subjects:
Brownian motion
Online Access:Get full text
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Summary: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 \(\phi(s) = s^{-k(d-1/H)}\).
ISSN:2331-8422