Mathematical Sciences: Some Problems in Stochastic Analysis

数学科学：随机分析中的一些问题

• 批准号: 8703333
• 负责人: Carl Mueller
• 金额: $ 3.08万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8703333&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Problems; Stochastic

The first part of this research will deal with Brownian motion on negatively curved manifolds. The investigator intends to find the Martin boundary for strongly negatively curved manifolds, and identify the bounded solutions of the heat equation. Also, conditions will be found under which harmonic measure on such manifolds is absolutely continuous with respect to angular measure, in polar coordinates. The second problem will deal with a nonlinear stochastic version of the heat equation. The investigator will study the rate at which the solution decays to zero. This will be a first step toward studying the qualitative behavior of such equations. The third problem will deal with the law of iterated logarithm for analytic functions, recently obtained by Markov. A version of Chung's law of the iterated logarithm will be sought for analytic functions. From this, packing dimension of the support of harmonic measure might be found in the same way as Markov found the Hausdorff dimension of the support.

本研究的第一部分将处理负弯曲流形上的布朗运动。 研究人员打算找到强负弯曲流形的马丁边界，并确定热方程的有界解。 此外，还将找到这样的条件：在极坐标中，此类流形上的谐波测量相对于角度测量绝对连续。 第二个问题将处理热方程的非线性随机版本。 研究人员将研究溶液衰减到零的速率。 这将是研究此类方程的定性行为的第一步。 第三个问题将涉及解析函数的迭代对数定律，该定律最近由马尔可夫获得。 将为解析函数寻找迭代对数钟定律的一个版本。 由此，调和测度支持的堆积维数可以用与马尔可夫发现支持的豪斯多夫维数相同的方式找到。