Open Access
April, 1989 On the Existence of Self-Intersections for Quasi-Every Brownian Path in Space
M. D. Penrose
Ann. Probab. 17(2): 482-502 (April, 1989). DOI: 10.1214/aop/1176991411


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.


Download Citation

M. D. Penrose. "On the Existence of Self-Intersections for Quasi-Every Brownian Path in Space." Ann. Probab. 17 (2) 482 - 502, April, 1989.


Published: April, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0714.60067
MathSciNet: MR985374
Digital Object Identifier: 10.1214/aop/1176991411

Primary: 60G17
Secondary: 60J55

Keywords: Brownian self-intersections , Hausdorff dimension , Local time , Ornstein-Uhlenbeck process on Wiener space

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 2 • April, 1989
Back to Top