On the slow points of fractional Brownian motion

TL;DR

This paper investigates the properties of slow points in fractional Brownian motion, introducing new methods to compute their Hausdorff dimension and extending the understanding of slow points in self-similar processes.

Contribution

It presents a novel approach for analyzing slow points in fractional Brownian motion, including new localization techniques and dimension calculations.

Findings

Established the Hausdorff dimension of fBm slow points

Developed a new method inspired by SPDEs for studying slow points

Extended the theory of slow points to 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.