Mathematical Sciences: Markov Processes

摘要我们研究由相互作用的粒子建模的复杂系统，或者更一般地，由具有高维状态空间的马尔可夫过程建模。在守恒定律存在的情况下，马尔可夫过程将有几个不变度量。如果相互作用是局域的，那么演化将把系统迅速带到由守恒量的局域平均值参数化的局域平衡。这些参数是空间和时间的函数，经过适当的重新标度后，会按照一定的演化规律演化。这种现象有几个例子，称为流体动力学标度：从粒子的哈密顿方程到流体的欧拉方程的转换，各种相互作用的粒子系统的整体扩散的非线性扩散方程等。该项目针对几类模型严格地研究了这种转换。在自然科学和社会科学中使用数学模型的核心问题之一是比例问题。一个是由大量相互作用的组件组成的复杂系统，该系统正在随时间演变。其属性是在单个组件级别指定的。然而，人们感兴趣的是，在大或全球范围内进行的某些少量测量是如何随着时间的推移而演变的。这项研究的目的是在某些特定的物理模型中建立两者之间的数学上有效的联系。

ABSTRACT We study complex systems modeled by interacting particles, or, more generally, by Markov Processes with a high dimensional state space. In the presence of conservation laws, the Markov Process will have several invariant measures. If the interaction is local then the evolution will take the system quickly to a local equilibrium parametrized by the local averages of the conserved quantities. These parameters that are functions of space and time will evolve according to some evolution law after suitable rescaling. There are several examples of this phenomenon, known as hydrodynamic scaling: the transition from Hamiltonian Equations for particles to Euler Equations for a fluid, Nonlinear diffusion equations for the bulk diffusion of various interacting particle systems etc. The project investigates rigorously this transition for several classes of models. One of the central problems in the use of mathematical models in the physical as well as social sciences, is the problem of scaling. One has a complex system consisting of a large number of interacting components which is evolving in time. Its properties are specified at the level of individual components. However one is interested in how certain small number of measurements made on the large or global scale evolve in time. The aim of this research is to make a mathematically valid link between the two in certain specific physical models.