Complex Stochastic Systems

随机过程的研究关注的是由“随机”或“机会”机制控制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。 要进行的研究涉及开发方法，指定这些数学模型，近似复杂的模型，由简单的，并构建模型，解决具体的科学应用。 本研究共分三个部分。 第一部分探讨了鞅特征的理论基础以及在非线性滤波中的应用，以及理解亚稳态物理现象的新方法。 第二部分的目的是发展新的方法来表示随机偏微分方程和测度值过程的大型聚集的“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。 第三部分的目标是系统地开发化学反应网络的随机模型，从经典的马尔可夫链模型开始，开发新的模型，考虑到涉及RNA和DNA分子的反应的逐步发展。拟议的研究是出于跨学科的问题，并解决物理和化学中的具体应用。 此外，要开发的理论和方法在其他领域，如计算机科学和金融的应用。 该项目还将为研究生和博士后研究人员提供肥沃的培训基地。 特别是，有一个训练有素的数学科学家的兴趣和必要的专业知识，以促进细胞和分子生物学中出现的问题的解决方案的高需求。

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 preformed is concerned with developing methods for specifying these mathematical models, approximating complex models by simpler ones, and constructing models addressing specific scientific applications. This research consists of three parts. The first part explores the theoretical foundations of the martingale characterization as well as applications to nonlinear filtering, and a new approach to understanding the physical phenomenon of metastability. The second part aims to develop new methods of representing stochastic partial differential equations and measure-valued processes in terms of large aggregations of "particles." These particle representations provide powerful tools for analyzing and approximating the associated models. The third part aims to systematically develop stochastic models for chemical reaction networks, beginning with classical Markov chain models and developing new models that take into account the stepwise development of reactions involving RNA and DNA molecules.The proposed research is motivated by interdisciplinary problems and addresses specific applications in physics and chemistry. In addition, the theory and methodology to be developed have applications in other areas such as computer science and finance. The project will also provide a fertile training ground for graduate students and postdoctoral researchers. In particular, there is a high demand for well-trained mathematical scientists with the interest and expertise necessary to contribute to the solution of problems arising in cell and molecular biology.