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 in 1908 and Perrin in 1909. The confirmation of Einstein's theory constituted empirical progress for the kinetic theory of heat. In essence, Einstein showed that the motion can be predicted directly from the kinetic model of thermal equilibrium. The importance of the theory lay in the fact that it confirmed the kinetic theory's account of the second law of thermodynamics as being an essentially statistical law. Smoluchowski's theory of Brownian motion starts from the same premise as that of Einstein and derives the same probability distribution "ρ"("x", "t") for the displacement of a Brownian particle along the "x" in time