2003/04/08 by L. R. G. Fontes, Luiz Renato Fontes, Marco Isopi +8
Computer Science · Decision Sciences · Mathematics · #60D05 #60F17 #60J65 #60K35 #82B41 #Cellular Automata and Applications #Evolutionary Algorithms and Applications #FOS: Mathematics #Innovation Diffusion and Forecasting #Probability (math.PR) #math.PR #msc:60D05 #msc:60F17 #msc:60J65 #msc:60K35 #msc:82B41
paper · pdf · doi:10.48550/arxiv.math/0304119
52 pages with 4 figures
arxiv created 2003/04/08 · openalex publication_date 2003/04/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Brownian Web (BW) is the random network formally consisting of the paths of coalescing one-dimensional Brownian motions starting from every space-time point in \mathbb R×\mathbb R. We extend the earlier work of Arratia and of Tóth and Werner by providing characterization and convergence results for the BW distribution, including convergence of the system of all coalescing random walkssktop/brownian web/finale/arXiv submits/bweb.tex 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) -- \em zero paths into the point from the past and exactly \em 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).