Coalescing Brownian flows: A new approach

TL;DR

This paper introduces a new framework for the coalescing Brownian flow, proving an optimal invariance principle with a simpler approach that extends to complex structures like the Sierpinski gasket.

Contribution

It provides a new state space and topology for coalescing Brownian flows and establishes an invariance principle under minimal assumptions, improving previous results.

Findings

New state space and topology for coalescing Brownian flows

Optimal invariance principle under finite variance assumption

Extension to coalescing flows on fractals like the Sierpinski gasket

Abstract

The coalescing Brownian flow on $\mathbb{R}$ is a process which was introduced by Arratia [Coalescing Brownian motions on the line (1979) Univ. Wisconsin, Madison] and T\'{o}th and Werner [Probab. Theory Related Fields 111 (1998) 375-452], and which formally corresponds to starting coalescing Brownian motions from every space-time point. We provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks. This result holds under a finite variance assumption and is thus optimal. In previous works by Fontes et al. [Ann. Probab. 32 (2004) 2857-2883], Newman et al. [Electron. J. Probab. 10 (2005) 21-60], the topology and state-space required a moment of order $3-\varepsilon$ for this convergence to hold. The proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work - in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the noncrossing property.