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Zbl 1105.60075
Fontes, L.R.G.; Isopi, M.; Newman, C.M.; Ravishankar, K.
The Brownian web: characterization and convergence. (English)
[J] Ann. Probab. 32, No. 4, 2857-2883 (2004). ISSN 0091-1798

The paper deals with the problem of characterization of the Brownian web, which is, rougly speaking, the collection of graphs of coalescing one-dimensional Brownian motions (with unit diffusion constant and zero drift) starting from all possible starting points in $1+1$-dimensional space-time. The authors consider the Brownian web as a random variable whose values are the collection of all paths from all possible starting points. Then they give a new characterization of the Brownian web in terms of a counting variable $\eta(t_0,t; a,b)$ which is the number of distinct points in $\Bbb R\times\{t_0+t\}$ which are touched by paths in the Brownian web which also touch some point in $[a,b]\times \{t_0\}$. Using this characterization the authors obtain a general theorem about the convergence to a Brownian web of a sequence of random variables under suitable conditions. As corollary they obtain, in a diffusive limit, the convergence of coalescing random walks to a Brownian web.
[Alessandro Pellegrinotti (Roma)]

MSC 2000:: *60K35 Interacting random processes
60J65 Brownian motion
60F17 Functional limit theorems
82B41 Random walks, etc. (statistical mechanics)
60D05 Geometric probability

Keywords: invariance principle; coalescing random walks; continuum limit

Cited in: Zbl 1143.82020 Zbl 1110.65010

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