Intermittency and Random Fractals

这项建议是关注的发展的分析/几何理论的随机场，主要是那些产生随机偏微分方程[SPDE]。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。 研究人员已经开发出基于几何测度理论的思想，用于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渐近分析的解决方案，一个大类的非线性SPDE。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。随机偏微分方程和随机场在理论数学、应用数学、数学海洋学、随机水文学、地质统计学、数学物理学和其他科学领域中发挥着重要作用。 刻画随机偏微分方程及相关随机场的精细局部和渐近结构是一个重要而又具有挑战性的课题。研究人员认为，该研究将具有足够的新奇，以解决随机偏微分方程和相关随机场理论中的一些长期存在的开放问题，并进一步促进其在各个科学领域的适用性。 此外，拟议的活动还将有助于培训研究生，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.