Brownian web 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 formula_1. The fact that the Brownian web consists of an "uncountable" number of Brownian motions is what makes the construction highly non-trivial. Arratia gave a construction but was unable to prove convergence of coalescing random walks to a limiting object and characterize such a limiting object. Tóth and Werner in their study of the true self-repelling motion obtained many detailed properties of this limiting object and its dual but did not prove convergence of coalescing
Brownian motion continuous stochastic process on a probability space (Ω, Σ, P) taking values in R. Then the following are equivalent: The infinitesimal generator (and hence characteristic operator) of a Brownian motion on R is easily calculated to be ½Δ, where Δ denotes the Laplace operator. In image processing and computer vision, the Laplacian operator has been used for various tasks such as blob and edge detection. This observation is useful in defining Brownian motion on an "m"-dimensional Riemannian manifold ("M", "g"): a Brownian motion on "M" is defined to be a diffusion on "M" whose characteristic operator formula_46 in local coordinates