Mathematical Sciences: Brownian Motion, Martingales and Applications

点击翻译按钮获取中文摘要

This research will focus on problems in two areas of martingales and their applications. First, the project will be study of laws of the iterated logarithm for harmonic functions and related questions. Second, the research will involve analysis of the inequalities for martingale transforms and projections of martingale transforms with applications to Riesz transforms and determination of the best constant in an inequality for the Haar functions. This research is in the general area of statistics and probability and consists of applying probabilistic techniques to problems in analysis. The work involves finding optimal or near optimal constants in a number of classical inequalities. Ten years ago such problems would have appeared unsolvable. Today the situation has changed completely and a number of important problems in several fields have been shown to reduce to the determination of certain constants in analytic inequalities.