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.
and random fields. These include asymptotic and local properties, potential theoryand geometry of the solutions of SPDEs and, more generally, random fields, andapproximations of the excursion probabilities of Gaussian random fields. The PIshave developed original methods for solving several outstanding open problems.Their discoveries have been published in more than thirty articles in peer reviewedjournals and presented at national and international conferences. The research ofthe Proposers has led to deeper understanding of SPDEs and random fields and will furtherpromote their applicability in various scientific areas such as mathematical physics, geostatistics, oceanography, cosmology, hydrology, just to mention a few. The past works of the Proposers demonstrate a strong commitment to theidentification and career developmentof young talent in the mathematical sciences. The PIs have integratednew discoveries in their research into courses they taught atthe University of Utah and Michigan State University, respectively. Professor Khoshnevisan has supervised 5 Ph.D. students [Dr. S.Y. Shiu (2011), Dr. L. Zhang (2012), Ms. K. Mac Arthur (current), P. Bezdek (current), and S. Li (current)], 3 post-doctoral scholars [Dr.s M. Joseph, K. Kim, and P. Mahboubi], 2 visiting scholars [Dr.s A. Ramos and L. Chen] and 2 Masters' students [Mr.s C. Robison and G. Bradway]. His 2 most-recent undergraduate research students have just enteredPhD programs: B. Chryst is now at Yale Statistics and D. Le Duc is at HarvardBiostatistics; the first is a female and the second is African-American.Both were funded in part by the Proposers' prior NSF grants. Professor Khoshnevisan has given 5 recent short courses [Seattle, August 2006; Taipei, June 2006; Aarhus, August 2007; Recife, August 2012; Michigan State University, 2013]. These courses are designed primarily for young researchers and advanced students. They are all based on the material of the [current and prior] NSF-funded research of the PIs. The lecture notes for the Recife course are available publicly, and are expected to be published by the Brazilian School of Probability in the near future as part of a compendium. The lectures notes for the NSF/CBMS course has been published by American Mathematical Society. Professor Xiao has supervised 5 Ph.D. students [Dr. Y. Xue (2011), Dr. W.-Y. Wu (2011), Dr. D. Cheng (2013), Y. Zhou (current) and A. Safikhani (current)] and mentoring 8 visiting scholars and post-doctoral scholars [Dr. Z. Chen, Ms. Y. Du, Dr. B. Li, Dr. J. Miao, Dr. M. A. Ouahra, Dr. T. Luks, Dr. X. Xiang, and Dr. L. Zhang]. He has given several graduate courses related to his researchat Michigan State University [2003, 2005, 2012], Marburg [Germany, Summer, 2005], Wuhan [China, July, 2009], Braunschweig[Germany, July 2010], Guangzhou [China, May 2012] and Hangzhou [China, June 2014]. All of the courses materials are based on the current and prior NSF grants. The PIs have helped to organize a number of activities relatedto the topics of their NSF-funded research. Those activities include a course on SPDEs for the CRM(2014); an NSF/CBMS conference (August, 2013), a conference in Banff on SPDEs (April 2012); the 36th Conference on Stochastic Processes & Their Applications in Boulder (2012); 6th Conference on L'evyProcesses &Their Applications in Dresden (2010); Seminar on StochasticProcesses (annual); Frontier Probability Days (regular). The Proposers areinvolved with editorial duties (Professor Xiao ischief probability editor of Statistics & Probability Letters and one of the two book editors for World Scientific Series on Probability Theory and Its Applications, both of PIs are on various editorial boards as Associate editors, and Professor Khoshnevisanis one of the two book editors for Probability & Its Applications as well as Progress in Probability).
