Special times for the point set process of the Brownian Net

TL;DR

This paper investigates the set of random times at which the point set process of the Brownian net loses its local finiteness, revealing that this set has Hausdorff dimension 1/2 almost surely, thus characterizing its fractal nature.

Contribution

It establishes that the set of times breaking local finiteness in the Brownian net's point process has Hausdorff dimension 1/2 almost surely, providing new fractal geometric insights.

Findings

The set of times breaking local finiteness has Hausdorff dimension 1/2.

The point set process is almost surely locally finite at deterministic times.

Random times where local finiteness fails form a fractal set with dimension 1/2.

Abstract

It is known that the point set process of the Brownian net is almost surely locally finite for all deterministic time, and there are random times that break this locally finiteness property. It is shown in this paper that the set of such random times has Hausdorff dimension 1/2 almost surely.