The Annals of Probability

Critical Brownian sheet does not have double points

Robert C. Dalang, Davar Khoshnevisan, Eulalia Nualart, Dongsheng Wu, and Yimin Xiao

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Abstract

We derive a decoupling formula for the Brownian sheet which has the following ready consequence: An $N$-parameter Brownian sheet in $\mathbf{R}^{d}$ has double points if and only if $d<4N$. In particular, in the critical case where $d=4N$, the Brownian sheet does not have double points. This answers an old problem in the folklore of the subject. We also discuss some of the geometric consequences of the mentioned decoupling, and establish a partial result concerning $k$-multiple points in the critical case $k(d-2N)=d$.

Article information

Source
Ann. Probab., Volume 40, Number 4 (2012), 1829-1859.

Dates
First available in Project Euclid: 4 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1341401150

Digital Object Identifier
doi:10.1214/11-AOP665

Mathematical Reviews number (MathSciNet)
MR2978539

Zentralblatt MATH identifier
1269.60053

Subjects
Primary: 60G60: Random fields
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60G15: Gaussian processes

Keywords
Brownian sheet multiple points capacity Hausdorff dimension

Citation

Dalang, Robert C.; Khoshnevisan, Davar; Nualart, Eulalia; Wu, Dongsheng; Xiao, Yimin. Critical Brownian sheet does not have double points. Ann. Probab. 40 (2012), no. 4, 1829--1859. doi:10.1214/11-AOP665. https://projecteuclid.org/euclid.aop/1341401150


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