Analysis and Geometry of Random Fields Related to Stochastic Partial Differential Equations and Random Matrices

Michigan State University, East Lansing MI

Investigators

Abstract

This project is concerned with the development of an analytic and geometric theory of random fields that arise from stochastic partial differential equations (SPDEs) and random matrices. Special emphasis is placed on vector-valued and matrix-valued random fields that play a central role in various areas of pure and applied mathematics, mathematical physics, astronomy, bio-imaging, mathematical oceanography, and statistics. The Principal Inestigator (PI) will develop probabilistic, analytic, and geometric tools for studying vector-valued and matrix-valued random fields that will lead to a deeper understanding of random fields that arise from systems of SPDEs and random matrices. These tools will have sufficient novelty to open new research areas, solve a number of open problems in the theory of SPDEs, random matrices, and related random fields. Moreover, the proposed activities will also help to train graduate students and to develop their careers in the mathematical and statistical sciences. It is significant and challenging to characterize the fine analytic and geometric structures of vector-valued and matrix-valued random fields. In the past investigations, the PI has established a series of results on Gaussian and, more generally, infinitely divisible random fields, and the solutions of SPDEs. Together with his collaborators, the PI has developed fractal geometry and potential theory for Gaussian random fields, additive Lévy processes, solutions to SPDEs, and used them to resolve several outstanding open problems in non-Markovian Gaussian and stable random fields, Lévy processes, and the theory of SPDEs. The PI plans to continue his investigation of precise quantitative connections between vector-valued and matrix-valued random fields, SPDEs, potential theory, and the geometry of random fractals. The proposed research will ultimately yield novel insights into the understanding of vector-valued and matrix-valued random fields, SPDEs, and random matrices. The expected results will not only contribute to the theories of random fields, SPDEs, and random matrices but also promote their applicability in mathematics, mathematical physics, and in other scientific areas. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.