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In probability theory, the Brownian web is an uncountable collection of one-dimensional coalescing Brownian motions, starting from every point in space and time. It arises as the diffusive space-time scaling limit of a collection of coalescing random walks, with one walk starting from each point of the integer lattice Z at each time. == History and Basic Description == What is now known as the Brownian web was first conceived by Arratia in his Ph.D. thesis and a subsequent incomplete and unpublished manuscript. Arratia studied the voter model, an interacting particle system that models the evolution of a population's political opinions. The individuals of the population are represented by the vertices of a graph, and each individual carries one of two possible opinions, represented as either 0 or 1. Independently at rate 1, each individual changes its opinion to that of a randomly chosen neighbor. The voter model is known to be dual to coalescing random walks (i.e., the random walks move independently when they are apart, and move as a single walk once they meet) in the sense that: each individual's opinion at any time can be traced backwards in time to an ancestor at time 0, and the joint genealogies of the opinions of different individuals at different times is a collection of coalescing random walks evolving backwards in time. In spatial dimension 1, coalescing random walks starting from a finite number of space-time points converge to a finite number of coalescing Brownian motions, if space-time is rescaled diffusively (i.e., each space-time point (x,t) gets mapped to (εx,ε^2t), with ε↓0). This is a consequence of Donsker's invariance principle. The less obvious question is: ''What is the diffusive scaling limit of the joint collection of one-dimensional coalescing random walks starting from'' every ''point in space-time?''Arratia set out to construct this limit, which is what we now call the Brownian web. Formally speaking, it is a collection of one-dimensional coalescing Brownian motions starting from every space-time point in . The fact that the Brownian web consists of an ''uncountable'' number of Brownian motions is what makes the construction highly non-trivial. The complete construction of the Brownian web was carried out later by Tóth and Werner in their study of the true self-repelling motion. The name Brownian web was coined by Fontes, Isopi, Newman and Ravishankar, where they introduced a topology for the Brownian web so that it is realized as a random variable taking values in a Polish space, in this case, the space of compact sets of paths. The introduction of this topology allows one to formalize mathematically the convergence of the coalescing random walks to the Brownian web. An extension of the Brownian web, called the Brownian net, has been introduced by Sun and Swart by allowing the coalescing Brownian motions to undergo branching. An alternative construction of the Brownian net was given by Newman, Ravishankar and Schertzer. For a recent survey, see Schertzer, Sun and Swart. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Brownian web」の詳細全文を読む スポンサード リンク
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