Zeros of the Brownian Sheet

TL;DR

This paper investigates the geometric properties of the zero set of the Brownian sheet, showing how projections affect its Hausdorff dimension and structure, thus answering an open problem in stochastic analysis.

Contribution

It proves that large-rank projections do not change the Hausdorff dimension of the zero set, and that lower-rank projections cover the entire projective space, resolving an open question.

Findings

Projections of sufficiently high rank do not alter the Hausdorff dimension of the zero set.

Lower-rank projections of the zero set cover the entire projective space.

Answers an open problem regarding the structure of zeros of the Brownian sheet.

Abstract

In this work we firstly answer to a question raised by Khoshnevisan in \cite[Open Problem 4]{khoshnevisan2007slices} by proving that almost surely there is no projection of big enough rank changing the Hausdorff dimension of the zeros of the Brownian sheet. Secondly, we prove that almost surely for every projection whose rank isn't matching the aforementioned condition, the projection of the zero set is the entirety of the projective space. Key words: Brownian sheet, zeros set, Hausdorff dimension, orthogonal projection.