Intermittency and Random Fractals

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This proposal is concerned with the development of an analytic/geometric theory of random fields, primarily those that arise from stochastic PDEs [SPDEs]. Special emphasis is placed on two extremal universality classes of SPDEs that are driven by fully non-linear multiplicative noise. The Investigators have developed ideas, based in geometric-measure theory, for the analysis of non-Markovian Gaussian and stable random fields. And they have introduced renewal-theoretic and coupling techniques for the asymptotic analysis of solutions to a large class of nonlinear SPDEs. They 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, physical multifractals, and related stochastic PDEs.Stochastic PDEs and random fields play a central role in various areas of pure and applied mathematics, mathematical oceanography, stochastic hydrology, geostatistics, mathematical physics and other scientific areas. It is significant and challenging to characterize the fine local and asymptotic structures of SPDEs and related random fields. The Investigators believe that the proposed research will have sufficient novelty to solve a number of long-standing open problems in the theory of stochastic PDEs and related random fields, and also further promote their applicability in various scientific areas. Moreover, the proposed activities will also help to train graduate students and to develop their careers in the mathematical and statistical sciences.