Complex Stochastic Systems

这个项目的第一部分集中在无限系统的随机方程。 这样的系统提供自然模型在自己的权利，但在这一部分的项目的结果也将提供必要的工具，随机偏微分方程的研究将被认为是在第二部分。 该项目的第二部分将开发方法来表示随机偏微分方程的解决方案的无限系统的“粒子”。“这些粒子表示为分析和近似相关模型提供了强大的工具。 在某些情况下，粒子表象自然产生于随机偏微分方程的唯象解释。 在其他情况下，它们只是提供了一个数学工具进行分析。 粒子系统通常被指定为随机微分方程的无限系统的解，并且在项目的第一部分中开发的唯一性结果将在获得所需的表示中发挥关键作用。该项目的第三部分继续研究细胞内反应网络的模型。一个目标将是提供算法，简化模型简化的早期工作的应用程序。 第二步是将早期的结果扩展到考虑细胞空间结构的模型中。随机过程的研究涉及对受“随机”或“偶然”机制支配的自然现象的数学描述。这种现象的数学模型可能试图描述时间、空间或两者的变化。要进行的研究是关注开发方法，指定这些数学模型，近似复杂的模型由简单的，并构建模型解决特定的科学应用。 一个基本的方法来指定模型的随机现象是制定方程的随机输入的解决方案是所需的模型。 不同的随机方程可以确定相同的模型，并且不同的方程可以提供对模型和模型所处理的现象的不同洞察。该项目关注的是如何开发和利用物理，生物和金融系统中自然出现的模型类别的多种表示。

The first part of this project focuses on infinite systems of stochastic equations. Such systems provide natural models in their own right, but results in this part of the project will also provide essential tools for the study of the stochastic partial differential equations to be considered in the second part. The second part of the project will develop methods for representing solutions of stochastic partial differential equation in terms of infinite systems of "particles." These particle representations provide powerful tools for analyzing and approximating the associated models. In some cases, the particle representations arise naturally from the phenomenological interpretation of the stochastic partial differential equation. In others, they simply provide a mathematical tool for its analysis. The particle systems are typically specified as solutions of infinite systems of stochastic differential equations, and uniqueness results to be developed in the first part of the project will play a critical role in obtaining the desired representations. The third part of the project continues work on models of intracellular reaction networks. One goal will be to provide algorithms that simplify the application of earlier work on model reduction. A second will be to extend the earlier results to models that take into account the spatial structure of the cell.The study of stochastic processes is concerned with mathematical descriptions of natural phenomena governed by "random" or "chance" mechanisms. Mathematical models of such phenomena may attempt to describe variation in time, in space, or both. The research to be performed is concerned with developing methods for specifying these mathematical models, approximating complex models by simpler ones, and constructing models addressing specific scientific applications. One fundamental approach to specifying models of stochastic phenomena is to formulate equations with stochastic inputs whose solution is the desired model. Different stochastic equations may determine the same model, and different equations may provide different insight into the model and the phenomenon the model addresses. The project is concerned with how to develop and exploit multiple representations for classes of models that arise naturally in physical, biological and financial systems.