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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