A cluster of sets of exceptional times of linear Brownian motion
type: Item
instance of: scholarly article
title: A cluster of sets of exceptional times of linear Brownian motion
zbMATH Open document ID: 1002.60077
author: Yasuki Isozaki
published in: Osaka Journal of Mathematics
publication date: 2003-01-01T00:00:00Z
review text: This paper considers weak maxima and points of increase of linear Brownian motion. To give an example of weak maxima, let \(V(\alpha ,x)=\text{sign}(x) |x |^\alpha\). Then \(t\) is a weak maximum if \[ \int_s^t V(\alpha_1,B_u-B_t) du \leq 0\;\forall s\in [0,t)\;\text{ and } \int V(\alpha_2, B_u-B_t) du \leq 0\;\forall s\in (t,1]. \] It is shown that the set of weak maxima in this sense has Hausdorff dimension \(1-(\rho_1+\rho_1)/2\) with positive probability, where \(\rho_i\) solves \(\sin \frac{\pi(1-\rho_i)}{2+\alpha_i} =\sin \frac{\pi\rho_i}{2+\alpha_i}.\) Different results of this type are presented, which all generalize a result of \textit{S. Aspadiiarov} and \textit{J-F. Le Gall} [Ann. Probab. 23, No. 4, 1605-1626 (1995; Zbl 0852.60091)] and use similar methods.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 1747463
zbMATH Keywords: Brownian motion
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: weak maxima
zbMATH Keywords: weak points of increase
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1747463