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，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等各个领域发挥着核心作用。这个研究项目是关于随机场的解析/几何理论的发展，主要是由SPDE产生的。该项目旨在开发概率、分析和几何工具，从而更深入地了解物理相关的随机场。预计这些工具将具有足够的新颖性，开辟新的研究领域，解决SPDE及相关随机场理论中一些长期存在的开放性问题，并进一步促进其适用性。这个项目有研究生参与研究。对SPDE及相关随机场的精细局部结构和渐近结构进行表征具有重要的意义和挑战性。在过去的工作中，研究人员发展了基于几何测量理论的思想，用于分析非马尔可夫高斯和稳定随机场，他们引入了更新理论和耦合技术，用于分析一类非线性SPDE的渐近解。该研究项目继续研究随机场、势理论、SPDE和随机分形几何之间的精确定量联系。特别强调了两个由完全非线性乘性噪声驱动的SPDE的极值通用性类。对这些联系的进一步研究有望对随机场的结构、物理多重分形和相关的SPDE产生新的见解。

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.