Critical Brownian sheet does not have double points

R. C. Dalang, D. Khoshnevisan, E. Nualart, D. Wu, and Y. Xiao

Abstract. We derive a decoupling formula for the Brownian sheet which has the following ready consequence: An N-parameter Brownian sheet in \({\bf R}^d\) has double points if and only if 2(d-2N)<d. In particular, in the critical case where 2(d-2N)=d, 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.

Keywords. Brownian sheet; multiple points; capacity; Hausdorff dimension.

AMS Classification (2000)


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Robert C. Dalang
Institut Math, EPFL
Station 8, CH-1015
Davar Khoshnevisan
Dept Math, U Utah
155 S 1400 E
Salt Lake City, UT 84112
Eulalia Nualart
Institut Galilée
U Paris 13
93430 Villetaneuse
Dongsheng Wu
Dept Math Sci
U Alabama-Huntsville
Huntsville, AL 35899
Yimin Xiao
Dept Stat & Probab
Michigan State U
East Lansing, MI 48824

Updates: September 13, 2010
© 2010- Robert C. Dalang, Davar Khoshnevisan, Eulalia Nualart, Dongsheng Wu, and Yimin Xiao