Multiple points of the Brownian sheet in critical dimensions

TL;DR

This paper investigates the existence of multiple points in the Brownian sheet at critical dimensions, establishing that such points almost surely do not exist when the parameters are at the critical threshold.

Contribution

It completes the characterization of multiple points in the Brownian sheet by proving their almost sure absence at critical dimensions.

Findings

No $k$-multiple points exist at critical dimensions $(k-1)d=2kN$

Confirms previous results for non-critical cases

Provides a complete picture of multiple points in Brownian sheets

Abstract

It is well known that an $N$-parameter $d$-dimensional Brownian sheet has no $k$-multiple points when $(k-1)d>2kN$, and does have such points when $(k-1)d<2kN$. We complete the study of the existence of $k$-multiple points by showing that in the critical cases where $(k-1)d=2kN$, there are a.s. no $k$-multiple points.