Random Fields and Potential Theory

9803747Khoshnevisan这笔拨款为犹他大学的Davar Khoshnevisan和Yimin Xiao提供资金，用于开发一种系统的方法来研究随机场研究中自然出现的许多特殊集。特别强调将放在高斯随机场，如布朗片和分数布朗运动。更具体地说，这些研究者将(1)研究随机场、容量和高阶偏微分方程之间的精确定量联系；(2)发展并推进了一类高斯随机场多重分形和几何分析的规范技术；(3)继续他们正在进行的工作，发展高斯过程理论与维纳空间容量之间的联系；(4)继续研究一类随机不稳定表面和结构上的正则热流。本研究将研究高斯随机场的性质，如布朗片和分数布朗运动。这些随机场在数学和经济学、海洋学、水文学和物理学的数学方面的几个学科中发挥着突出的作用。提出的研究目标是开发分析工具，以更好地理解随机场的几何问题，并有助于促进其进一步的适用性。

9803747KhoshnevisanThis grant provides funds for Davar Khoshnevisan and Yimin Xiao at the University of Utah to develop a systematic approach to the study of many of the exceptional sets which naturally arise in the study of random fields. Special emphasis will be placed on Gaussian random fields such as the Brownian sheet and fractional Brownian motion. More specifically, these investigators will (1) study precise quantitative connections between random fields, capacities and higher--order partial differential equations; (2) develop and advance canonical techniques for the multi-fractal and geometric analysis of a large class of Gaussian random fields; (3) continue their on-going work on developing connections between the theory of Gaussian processes and capacity in Wiener space; and (4) continue their investigation of canonical heat flow on a class of random erratic surfaces and structures.This research will study the properties of Gaussian random fields, such as the Brownian sheet and fractional Brownian motion. These are random fields which play a prominent role in several disciplines in mathematics and mathematical aspects of economics, oceanography, hydrology and physics. The goal of the proposed research is to develop analytical tools which will lead to a better understanding of geometric problems for random fields and help promote their further applicability.