Collaborative Research: Fractals, Multifractals, and Stochastic Partial Differential Equations

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Stochastic partial differential equations (SPDE) are employed to model a wide range of natural phenomena and play a central role in various areas of pure and applied mathematics, mathematical oceanography, stochastic hydrology, geo-statistics, and mathematical physics. This research project is concerned with the development of an analytic/geometric theory of random fields, primarily those that arise from SPDE. The project aims to develop probabilistic, analytic, and geometric tools that will lead to a deeper understanding of physically-relevant random fields. It is anticipated that these tools will have sufficient novelty to open new research areas, solve a number of long-standing open problems in the theory of SPDE and related random fields, and further promote their applicability. The project involves graduate students in the research.It is significant and challenging to characterize the fine local and asymptotic structures of SPDE and related random fields. In past work, the investigators developed ideas, based in geometric-measure theory, for the analysis of non-Markovian Gaussian and stable random fields, and they introduced renewal-theoretic and coupling techniques for the asymptotic analysis of solutions to a large class of nonlinear SPDE. This research project continues investigation of precise quantitative connections between random fields, potential theory, SPDE, and the geometry of random fractals. Special emphasis is placed on two extremal universality classes of SPDE that are driven by fully non-linear multiplicative noise. Further pursuit of these connections is expected to yield novel insights into the structure of random fields, physical multifractals, and related SPDE.