This 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.
