The Brownian Web: Characterization and Convergence

The Brownian Web (BW) is the random network formally consist- ing of the paths of coalescing one-dimensional Brownian motions start- ing from every space-time point in R×R. We extend the earlier work of Arratia and of Toth and Werner by providing characterization and convergence results for the BW distribution, including convergence of the system of all coalescing random walks to the BW under diffusive space-time scaling. We also provide characterization and convergence results for the Double Brownian Web, which combines the BW with its dual process of coalescing Brownian motions moving backwards in time, with forward and backward paths "reflecting" off each other. For the BW, deterministic space-time points are almost surely of "type" (0,1) — zero paths into the point from the past and exactly one path out of the point to the future; we determine the Hausdorff dimension for all types that actually occur: dimension 2 for type (0,1), 3/2 for (1,1) and (0,2), 1 for (1,2), and 0 for (2,1) and (0,3).

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