Markov Processes

在这个项目中，研究人员将研究具有守恒量的复杂相互作用系统的行为。它们的演化通常是在微观层面上描述的，而问题则是在更大或更宏观的尺度上提出的。从一个尺度到另一个尺度的转变总是需要在微观层面上进行一些平均，有时，就像在非梯度模型中那样，也需要消除某些分歧。在双曲标度的情况下，宏观方程是一个发展激波的非线性双曲方程。激波微观轮廓的表征是一个有趣的问题。研究单个标记粒子与大量粒子相互作用的重标度运动，需要在非常小的混合假设下证明一个中心极限定理。许多复杂的物理系统由大量的基本分量组成。这些组件中的每一个都只与几个相邻的组件以某种方式交互。然而，大系统作为一个整体，由于各个组件的相互作用，以某种复杂的方式发展。这个项目致力于通过几个具体的例子来探索这一现象。其目标是提供一种机制，用于基于对小范围内相互作用的性质的假设来对大范围内的行为进行推断。

9803140VaradhanIn this project the investigator will study the behavior of complex interactive systems with conserved quantities. Their evolution is often described at the microscopic level while questions are asked at a much larger or macroscopic scale. The transition from one scale to another always requires some averaging at the microscopic level and sometimes, as in the non gradient models, getting rid of certain divergences as well. In the case of hyperbolic scaling, the macroscopic equation is a nonlinear hyperbolic equation that develops shocks. The characterization of the microscopic profile of the shock is an interesting problem. Studying the rescaled motion of a single tagged particle that interacts with large number of particles requires a central limit theorem to be proved under very minimal mixing assumptions.Many complex physical systems consist of a large number of elementary components. Each of these components interacts in some fashion only with a few neighboring ones. However, the large system, as a whole, evolves in some complex manner due to the interactions of the individual components. This project is devoted to the exploration of this phenomenon, in several concrete examples. The goal is to provide mechanisms for making inferences about the behavior in the large scale based on assumptions on the nature of the interaction in the small scale.