The Annals of Probability

On the Existence of Self-Intersections for Quasi-Every Brownian Path in Space

M. D. Penrose

Full-text: Open access

Abstract

The set of self-intersections of a Brownian path $b(t)$ taking values in $\mathbb{R}^3$ has Hausdorff dimension 1, for almost every such path, with respect to Wiener measure, a result due to Fristedt. Here we prove that this result (together with the corresponding result for paths in $\mathbb{R}^2$) in fact holds for quasi-every path with respect to the infinite-dimensional Ornstein-Uhlenbeck process, a diffusion process on Wiener space whose stationary measure is Wiener measure. We do this using Rosen's self-intersection local time, first proving that this exists for quasi-every path.

Article information

Source
Ann. Probab., Volume 17, Number 2 (1989), 482-502.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991411

Digital Object Identifier
doi:10.1214/aop/1176991411

Mathematical Reviews number (MathSciNet)
MR985374

Zentralblatt MATH identifier
0714.60067

JSTOR
links.jstor.org

Subjects
Primary: 60G17: Sample path properties
Secondary: 60J55: Local time and additive functionals

Keywords
Brownian self-intersections Hausdorff dimension local time Ornstein-Uhlenbeck process on Wiener space

Citation

Penrose, M. D. On the Existence of Self-Intersections for Quasi-Every Brownian Path in Space. Ann. Probab. 17 (1989), no. 2, 482--502. doi:10.1214/aop/1176991411. https://projecteuclid.org/euclid.aop/1176991411


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