Mathematical Sciences: Studies in Brownian Motion and Random Walk

点击翻译按钮获取中文摘要

9626642 Lawler ABSTRACT The proposer will continue investigation of the relationship between critical exponents for Brownian motion and random walk and geometric properties of paths in two and three dimensions. It is hoped to establish the existence of a nontrivial multifractal spectrum for harmonic measure of a Brownian path, where the spectrum is given in terms of certain critical exponents for Brownian motion. The proposer will also continue the program of investigating nonintersecting Brownian motions with the aim to establish rigorously the predictions of two-dimensional exponents given by nonrigorous conformal field theory. Random walk analogues will be studied at the same time. Other geometric properties of paths will be studied, such as the percolation exponent for Brownian motion which quantifies the notion of the thinnest subpath. Study of the percolation exponent will lead to the study of processes obtained by removing loops from the Brownian path. The path of a Brownian motion is an example of a random multifractal, a set whose ``dimension'' looks different at different points. It is the path traversed by a randomly moving particle. Multifractals have been studied in a number of contexts in science, but there are very few examples where one can establish rigorously the multifractal behavior. The proposer intends to try to analyze rigorously the multifractal behavior for the Brownian motion path. Related to this project is the attempt to understand ``nonconformal field theory''---a powerful tool in theoretical physics which is not sufficiently understood on a mathematical level to allow for rigorous analysis.