Mathematical Sciences: Studies in Brownian Motion and Random Walk

摘要本文将继续研究布朗运动与随机游走的临界指数之间的关系以及二维和三维路径的几何性质。希望建立布朗路径谐波测度的非平凡多重分形谱的存在性，其中谱用布朗运动的某些临界指数给出。提议者还将继续研究非相交布朗运动的计划，目的是严格地建立由非严格共形场论给出的二维指数的预测。与此同时，我们将研究随机漫步的类似物。路径的其他几何性质将被研究，例如布朗运动的渗透指数，它量化了最薄子路径的概念。对渗透指数的研究将导致对从布朗路径中去除环路所得到的过程的研究。布朗运动的路径是随机多重分形的一个例子，它的“维数”在不同的点上看起来不同。它是一个随机移动的粒子所经过的路径。多重分形已经在许多科学背景下进行了研究，但很少有例子可以严格地建立多重分形行为。作者试图对布朗运动路径的多重分形行为进行严格的分析。与这个项目相关的是试图理解“非共形场论”——理论物理学中的一个强大工具，但在数学层面上还没有得到充分的理解，无法进行严格的分析。

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.