Critical Brownian sheet does not have double points

TL;DR

This paper proves that the critical Brownian sheet in -dimensional space with 4 parameters does not have double points, resolving a longstanding open problem and exploring related geometric properties.

Contribution

It establishes a decoupling formula for the Brownian sheet and determines the absence of double points in the critical case where the dimension equals four times the number of parameters.

Findings

Brownian sheet has double points iff d<4N

In the critical case d=4N, no double points occur

Partial results on k-multiple points in critical cases

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 $k(d-2N)=d$.