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On the slow points of fractional Brownian motion

Davar Khoshnevisan, Cheuk Yin Lee

Abstract

Esser and Loosveldt have recently resolved a long-standing open problem in the folklore by proving that fractional Brownian motion (fBm) has slow points in the sense of Kahane, following a rich theory of slow points developed for Brownian motion and other, related, self-similar Markov processes. We presently introduce another method for the study of slow points in order to compute the Hausdorff dimension of fBm slow points. Our method follows recent ideas on the points of slow growth for SPDEs but also requires a number of new localization ideas that are likely to have other applications.

On the slow points of fractional Brownian motion

Abstract

Esser and Loosveldt have recently resolved a long-standing open problem in the folklore by proving that fractional Brownian motion (fBm) has slow points in the sense of Kahane, following a rich theory of slow points developed for Brownian motion and other, related, self-similar Markov processes. We presently introduce another method for the study of slow points in order to compute the Hausdorff dimension of fBm slow points. Our method follows recent ideas on the points of slow growth for SPDEs but also requires a number of new localization ideas that are likely to have other applications.
Paper Structure (5 sections, 9 theorems, 82 equations)

This paper contains 5 sections, 9 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Localization
  3. Proof of Theorem \ref{['th:main']}
  4. Proof of the upper bound
  5. Proof of the lower bound

Key Result

Theorem 1.1

Choose and fix a compact set $K \subset (0\,,\infty)$. Then, almost surely, where $\mathop{\mathrm{\underline{dim}_{_{\rm M}}\!}}\nolimits$ and $\mathop{\mathrm{dim_{_{\rm H}}}}\nolimits$ respectively denote the lower Minkowski dimension and the Hausdorff dimension Falconer.

Theorems & Definitions (16)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 6 more