Mathematical Sciences: The Brownian Sheet and Related Processes

9503290 Khoshnevisan摘要N参数布朗单是一个平均零高斯过程，它是对N维矩形轴上两侧的标准白噪声进行积分得到的。这位研究人员和他的同事提议调查这一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组的研究中产生影响。还将考虑被视为非光滑随机曲面的布朗单的图的性质。感兴趣的问题包括某些问题，涉及一些随机的不规则裂缝和表面上的运输问题。这些物体的理想化版本出现在最近的工程和物理文献中的研究中。拟议的研究涉及对一类随机过程的系统研究，这些随机过程出现在几个数学领域以及物理和工程中。这些问题包括但不限于运输问题、波浪现象和湍流。在许多这样的问题中，潜在的共同主题是一个称为布朗单的数学对象。除了少数例外，布朗单的几何在很大程度上是一个未被探索的领域；主要是因为没有太多的一般方法来研究它。在其他工作中，这项提议的目标是做出一些尝试来纠正这种情况。

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.