Mathematical Sciences: "Percolation on Trees, Intersections of Random Sets, and Measures of Full Hausdorff Dimension"

点击翻译按钮获取中文摘要

Peres The proposer intends to study several problems in Probability theory, concerning tree-indexed processes, intersections of "small" random sets, and Hausdorff dimension. The problems involve potential theory for images of "small" sets under Brownian motion, and the relation between branching processes and ranges of certain lattice random walks. Intersections of Brownian sample paths were first studied by Dvoretzky, Erdos and Kakutani more than forty years ago. Intersections of random walk paths are in some ways harder to analyze; Lawler's (1991) book presents most of what is known about them. The proposer's approach to these problems is motivated by his recent work, which shows that certain sample paths (e.g. of Brownian motion ) are "equivalent" to certain "random Cantor sets" constructed via a branching process, in the sense that the same deterministic sets are hit by these two types of random sets with positive probability. The remaining problems have close ties to ergodic theory; they involve the Hausdorff dimension of measures on Galton Watson trees and on invariant sets for expanding maps. It is proposed to study several problems in probability theory involving intersections of random paths in space. Some of these problems were studied by mathematicians forty years ago, but in the last decade there has been renewed interest, as mathematical physicists have shown the relevance of this topic to questions of physical interest. Classical notions of "Fractional dimension" play an important role in this study. A recurring theme is that complicated configurations are best understood by finding the most uniform way to distribute mass on them. New mathematical tools, developed for very different purposes, offer an exciting approach to these problems.