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Critical Brownian sheet does not have double points
Robert C. Dalang, Davar Khoshnevisan, Eulalia Nualart, Dongsheng Wu, and Yimin Xiao
Source: Ann. Probab. Volume 40, Number 4
(2012), 1829-1859.
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 $k(d-2N)=d$.
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1341401150
Digital Object Identifier: doi:10.1214/11-AOP665
Zentralblatt MATH identifier: 06067457
Mathematical Reviews number (MathSciNet): MR2978539
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