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July 2012 Critical Brownian sheet does not have double points
Robert C. Dalang, Davar Khoshnevisan, Eulalia Nualart, Dongsheng Wu, Yimin Xiao
Ann. Probab. 40(4): 1829-1859 (July 2012). DOI: 10.1214/11-AOP665


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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Robert C. Dalang. Davar Khoshnevisan. Eulalia Nualart. Dongsheng Wu. Yimin Xiao. "Critical Brownian sheet does not have double points." Ann. Probab. 40 (4) 1829 - 1859, July 2012.


Published: July 2012
First available in Project Euclid: 4 July 2012

zbMATH: 1269.60053
MathSciNet: MR2978539
Digital Object Identifier: 10.1214/11-AOP665

Primary: 60G60
Secondary: 60G15 , 60J45

Keywords: Brownian sheet , capacity , Hausdorff dimension , multiple points

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.40 • No. 4 • July 2012
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