Intermittency and Random Fractals

本提案关注随机场的解析/几何理论的发展，主要是那些来自随机偏微分方程[SPDEs]。特别强调了由完全非线性乘性噪声驱动的两类spde的极值通用性。研究人员在几何测量理论的基础上发展了分析非马尔可夫高斯随机场和稳定随机场的思想。他们还引入了更新理论和耦合技术，用于求解一类非线性spde的渐近分析。他们计划继续研究随机场、势理论、随机偏微分方程和随机分形几何之间的精确定量联系。他们相信，对这些联系的进一步研究将最终对随机场的结构、物理多重分形和相关的随机偏微分方程产生新的见解。随机偏微分方程和随机场在纯数学和应用数学、数学海洋学、随机水文学、地质统计学、数学物理和其他科学领域的各个领域发挥着核心作用。对SPDEs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。研究者认为，本研究将具有足够的新颖性，解决随机偏微分方程及相关随机场理论中一些长期存在的开放性问题，并进一步促进其在各个科学领域的应用。此外，拟议的活动也将有助于训练研究生，并发展他们在数学和统计科学方面的职业。

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.