Special points of the Brownian net

TL;DR

This paper provides a detailed structural analysis of the Brownian net, a stochastic process that generalizes the Brownian web by incorporating branching, including a classification of points based on path configurations.

Contribution

It offers an almost sure classification of points in the Brownian net and explores its structural properties, advancing understanding of this stochastic process.

Findings

Almost sure classification of points in the Brownian net

Structural properties of the Brownian net established

Enhanced understanding of path configurations in the Brownian net

Abstract

The Brownian net, which has recently been introduced by Sun and Swart [SS08], and independently by Newman, Ravishankar and Schertzer [NRS08], generalizes the Brownian web by allowing branching. In this paper, we study the structure of the Brownian net in more detail. In particular, we give an almost sure classification of each point in $R^2$ according to the configuration of the Brownian net paths entering and leaving the point. Along the way, we establish various other structural properties of the Brownian net.