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The Brownian web: characterization and convergence. (English) Zbl 1105.60075

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 \(\mathbb R\times\{t_0+t\}\) 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J65 Brownian motion
60F17 Functional limit theorems; invariance principles
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60D05 Geometric probability and stochastic geometry
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