Diffusion Processes in Composite Media and Related Problems

9625642 Zheng摘要 研究人员将继续研究与复合介质中热传播有关的扩散过程。建议的研究涉及概率论的几个领域：狄利克雷形式和马尔可夫过程理论，弱收敛理论，随机微积分和鞅理论。研究者主要关注以下问题： 1)含时Dirichlet形式与动边界问题 2)分析了给出上述问题近似解的马尔可夫链模型; 3)马氏链序列逼近扩散过程时弱收敛速度的估计。研究者期望本研究能为扩散过程理论的现有结果提供更多的应用，从而丰富整个理论。 本研究的主要目的是研究由几种不同材料组成的物体的温度演化问题。两个例子是海洋上冰山的融化-冻结过程和核反应堆内的温度控制。对于这些问题，真实的计算是非常复杂的，有时应用科学家甚至不知道他们的计算结果是否足够精确到适用于真实的世界。 研究人员将使用一种称为"扩散过程"的特殊数学模型来模拟这些物体上的真实的温度演变。此外，研究人员将研究如何在计算机上实现这一过程，以及计算机模拟结果的准确性。

9625642 Zheng ABSTRACT The investigator will continue the study of diffusion process related to heat propagation in a composite medium. The proposed research is related to several areas of probability theory: the theory of Dirichlet forms and Markov processes, weak convergence theory, stochastic calculus and martingale theory. The investigator is mainly interested in the following problems: 1) Time-dependent Dirichlet form and moving boundary problems; 2) The analysis of the Markov chain models which give approximating solutions for the above problems; 3) The estimate of the rate of weak convergence when a sequence of Markov chains approach a diffusion process. The investigator expects that this proposed study will introduce more applications of the existing results of diffusion process theory thereby enriching the whole theory. The main purpose of this investigation is to study problems related to the temperature evolution of an object composed of several different materials. Two examples are the melting-freezing process of ice hills on the oceans and temperature control inside a nuclear reactor. For those problems, the real computations are very complicated and sometimes applied scientists do not even know if their computational result is precise enough to the applicable to the real world. The investigator will use a special mathematical model known as "diffusion processes" to simulate the real temperature evolution on those objects. Moreover, the investigator will study how to realize this process on a computer and how accurate the computer simulation result will be.