Geometry of Random Fields and Stochastic Partial Differential Equations

University Of Utah, Salt Lake City UT

Investigators

Abstract

This proposal is concerned with the development of a systematic approach to the study of the analytic and geometric properties of random fields and stochastic partial differential equations (SPDEs). Special emphasis is placed on Gaussian, stable, and Levy random fields such as the Brownian sheet and additive Levy processes, as well as the solutions of SPDEs that are driven by Gaussian or Levy noises. The mentioned examples are random fields that arise naturally in various areas of pure and applied mathematics, mathematical oceanography, stochastic hydrology, geostatistics, and mathematical as well as statistical physics. The proposed research plans to gather and develop probabilistic, analytic, and geometric tools that will lead to a deeper understanding of the analysis and geometry of various random fields. The Proposers believe that these tools will have sufficient novelty to solve a number of long-standing open problems in the theory of random fields, and also further promote their further applicability. In their past investigations, the Proposers have developed potential theories for additive Levy processes and the Brownian sheet, and used them to resolve several outstanding open problems in the theory of Levy processes and the analysis of the Brownian sheet. The Proposers have developed ideas, based in geometric-measure theory, for investigating non-Markovian Gaussian and stable random fields. And they have introduced renewal-theoretic techniques for the asymptotic analysis of solutions to a large class of parabolic stochastic PDEs driven by singular random noises. The Proposers plan to continue their investigation of precise quantitative connections between random fields, potential theory, stochastic PDEs, and the geometry of random fractals. And they believe that further pursuit of these connections will ultimately yield novel insights into the structure of random fields and related stochastic PDEs.