TY - JOUR

T1 - Coalescing Brownian flows

T2 - A new approach

AU - Berestycki, Nathanaël

AU - Garban, Christophe

AU - Sen, Arnab

N1 - Publisher Copyright:
© Institute of Mathematical Statistics, 2015.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

N2 - The coalescing Brownian flow on R is a process which was introduced by Arratia [Coalescing Brownian motions on the line (1979) Univ.Wisconsin, Madison] and Tóth 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 - ε 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.

AB - The coalescing Brownian flow on R is a process which was introduced by Arratia [Coalescing Brownian motions on the line (1979) Univ.Wisconsin, Madison] and Tóth 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 - ε 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.

KW - Arratia flow

KW - Brownian web

KW - Coalescing Brownian motions

KW - Coalescing flow on Sierpinski gasket

KW - Coalescing random walks

KW - Invariance principle for coalescing random walks

KW - Non-crossing property

KW - Schramm-Smirnov space of coalescing flows

UR - http://www.scopus.com/inward/record.url?scp=84951043467&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84951043467&partnerID=8YFLogxK

U2 - 10.1214/14-AOP957

DO - 10.1214/14-AOP957

M3 - Article

AN - SCOPUS:84951043467

VL - 43

SP - 3177

EP - 3215

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 6

ER -