The Brownian web is a two-dimensional black noise

TL;DR

This paper proves that the Brownian web, a collection of coalescing Brownian motions starting from every point on the plane, is a two-dimensional black noise, confirming a long-standing theoretical question.

Contribution

It provides the first rigorous proof that the Brownian web is a two-dimensional black noise, expanding the class of known examples beyond critical planar percolation.

Findings

Brownian web is a two-dimensional black noise

Confirms the Brownian web's sensitivity to small-area resampling

Expands understanding of two-dimensional noise models

Abstract

The Brownian web is a random variable consisting of a Brownian motion starting from each space-time point on the plane. These are independent until they hit each other, at which point they coalesce. Tsirelson mentions this model in his paper "Scaling limit, Noise, Stability", along with planar percolation, in suggesting the existence of a two-dimensional black noise. A two-dimensional noise is, roughly speaking, a random object on the plane whose distribution is translation invariant and whose behavior on disjoint subsets is independent. Black means sensitive to the resampling of sets of arbitrarily small total area. Tsirelson implicitly asks: "Is the Brownian web a two-dimensional black noise?". We give a positive answer to this question, providing the second known example of such after the scaling limit of critical planar percolation.