## Preprint: 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)

Support.

• The research of R.C.D. was supported in part by a grant from the Swiss National Foundation for Scientific Research.
• The research of D.K. and Y.X. was supported in part by grants from the United States National Science Foundation.
Pre/E-Prints. This paper is available in

 Robert C. Dalang Institut Math, EPFL Station 8, CH-1015 Lausanne Switzerland robert.dalang@epfl.ch Davar Khoshnevisan Dept Math, U Utah 155 S 1400 E Salt Lake City, UT 84112 U.S.A. davar@math.utah.edu Eulalia Nualart Institut Galilée U Paris 13 93430 Villetaneuse France nualart@math.univ-paris13.fr Dongsheng Wu Dept Math Sci U Alabama-Huntsville Huntsville, AL 35899 U.S.A. Dongsheng.Wu@uah.edu Yimin Xiao Dept Stat & Probab Michigan State U East Lansing, MI 48824 U.S.A. xiao@stt.msu.edu