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

随机偏微分方程（SPDE）被用来模拟广泛的自然现象，并在纯数学和应用数学、数学海洋学、随机水文学、地质统计学和数学物理学的各个领域发挥着核心作用。本研究项目关注的是随机场的分析/几何理论的发展，主要是那些从SPDE产生的。 该项目旨在开发概率，分析和几何工具，从而加深对物理相关随机场的理解。 预计这些工具将具有足够的新奇，以开辟新的研究领域，解决一些长期存在的开放问题的理论SPDE和相关的随机场，并进一步促进其适用性。本课题的研究对象是研究生，对随机偏微分方程及相关随机场的精细局部结构和渐近结构的刻画具有重要意义和挑战性。 在过去的工作中，研究人员开发的想法，在几何测度理论的基础上，分析非马尔可夫高斯和稳定的随机场，他们介绍了更新理论和耦合技术的解决方案的渐近分析一大类非线性SPDE。这个研究项目继续调查随机场，势理论，SPDE和随机分形几何之间的精确定量联系。特别强调的是放在两个极端的普遍性类的SPDE是由完全非线性乘性噪声驱动。 这些连接的进一步追求有望产生新的见解随机场的结构，物理多重分形，和相关的SPDE。

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.