Random Fields and Stochastic Partial Differential Equations

这一建议的主要问题之一是系统地发展分析随机场和随机微分方程的几何方法。特别强调了高斯场、稳定场和列维随机场，如布朗单、加性列维过程和由高斯噪声或列维噪声驱动的随机偏微分方程解。这些是在数学、应用数学、数学海洋学、湍流、随机水文学、地质统计学、空间统计学和数学物理的许多领域中自然产生的随机场。这项拟议的研究旨在收集和开发概率、解析、代数和几何工具，以加深对随机场和随机偏微分方程几何的理解。提出者相信，这些工具将足够新颖，可以解决随机领域中的一些公开问题。提出者还认为，本文开发的程序将有助于促进这些方法和相关数学方法在上述领域的未来适用性。

One of the primary concerns of this proposal is to develop, systematically, geometric methods for analysisof random fields and stochastic differential equations. Special emphasis isplaced on Gaussian, stable, and Levy random fields such as theBrownian sheet, additive Levy processes, and solutions ofstochastic partial differential equations driven by Gaussian orLevy noises. These are random fields that arise naturally inmany areas of mathematics, applied mathematics, mathematicaloceanography, turbulence, stochastic hydrology, geostatistics, spatial statistics, and mathematical physics. The proposed research intends to gather and develop probabilistic, analytic, algebraic, and geometric tools that lead to a deeper understanding of the geometry of random fields and stochastic partial differential equations. The Proposers believe that these tools will be novel enough to solve a number of open problems on random fields. The Proposers also believe thatthe program developed herein will help promote the future applicability of these and related mathematical methods in theaforementioned areas.