On the existence of self-intersections for quasi-every Brownian path in space
type: Item
instance of: scholarly article
title: On the existence of self-intersections for quasi-every Brownian path in space
zbMATH Open document ID: 0714.60067
author: Mathew D. Penrose
published in: The Annals of Probability
publication date: 1989-01-01T00:00:00Z
review text: \textit{M. Fukushima} [J. Math. Soc. Japan 36, No.1, 161-176 (1984; Zbl 0522.60081)] defined the notion of quasi-everywhere (q.e.) for standard Brownian motion. The main purpose of the present paper is to show that much of the theory due to \textit{J. Rosen} [Commun. Math. Phys. 88, No.3, 327-338 (1983; Zbl 0534.60070)] on the existence and properties of the self-intersection of almost every Brownian path carries over to results for quasi-every Brownian path. In \(R^ 3\) it is not only shown that quasi-every Brownian path intersects itself in any time interval, but also that the set of double points has Hausdorff dimension 1, thereby strengthening the a.e. result due to \textit{B. Fristedt} [Z. Wahrscheinlichkeitstheorie Verw. Gebiete 9, 62-64 (1967; Zbl 0314.60030)]. In order to show that the self-intersection local time on a bounded time set is q.e. positive, the author requires an upper estimate of the small tail of the distribution of the intersection local time of two independent Brownian motions starting at the same point. The reviewer would like to know the asymptotic behavior of this small tail.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 4176158
zbMATH Keywords: Brownian motion
zbMATH Keywords: self-intersection
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: self- intersection local time
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991411
OpenAlex ID: W1987273957
DOI: 10.1214/AOP/1176991411
description: scientific article; zbMATH DE number 4176158