Mathematical Sciences: The Brownian Sheet and Related Processes

数学科学：布朗表及相关过程

• 批准号: 9503290
• 负责人: Davar Khoshnevisan
• 金额: $ 2万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9503290&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Brownian; Sheet; Related

9503290 Khoshnevisan Abstract The N-parameter Brownian sheet is a mean zero Gaussian process which is obtained by integrating standard white noise on N-dimensional rectangles with two sides on the axes. The investigator and his colleagues propose to investigate some of the probabilistic and analytical properties of this process. This process locally exhibits many features of the solutions to a class of stochastic partial differential equations. As such, our proposed analysis would have ramifications in the study of hyperbolic stochastic partial differential equations. Also considered will be the properties of the graph of the Brownian sheet viewed as a non-smooth random surface. The problems of interest include certain questions involving transport problems on some random erratic fractures and surfaces. Idealized versions of such objects appear in recent investigations in the engineering and physics literature. The proposed research involves a systematic study of a class of random processes which arise in several areas of mathematics as well as in physics and engineering. Such problems include but are not limited to transport problems, wave phenomena and turbulence. The underlying common theme in many such problems is a mathematical object called the Brownian sheet. With only a few exceptions, the geometry of the Brownian sheet is a largely unexplored area; primarily due to the fact that there are not many general methods for its study. Among other works, it is the goal of this proposal to make some attempts at remedying this situation.

9503290 Khoshnevisan摘要 N-参数布朗单是一个均值为零的高斯过程，它是通过对标准白色噪声在N维矩形上积分得到的，矩形的两条边都在轴上。 研究者和他的同事们建议研究这个过程的一些概率和分析性质。这一过程局部地表现出一类随机偏微分方程解的许多特征。因此，我们提出的分析将在双曲型随机偏微分方程的研究中产生影响。也将考虑被视为一个非光滑的随机表面的布朗单的图形的属性。 感兴趣的问题包括某些问题，涉及运输问题的一些随机不规则的裂缝和表面。这种物体的理想化版本出现在最近的工程和物理学文献中。 拟议的研究涉及一类随机过程的系统研究，这些过程出现在数学、物理学和工程学的几个领域。这些问题包括但不限于运输问题、波动现象和湍流。在许多这样的问题中，潜在的共同主题是一个被称为布朗单的数学对象。 除了少数例外，布朗单的几何是一个很大程度上未探索的领域;主要是因为没有太多的通用方法来研究它。除其他工作外，本建议的目标是为纠正这种情况作出一些尝试。