Research in Stochastic Processes and Nonlinear Filtering

拉利研究将在两个不同的领域进行：（1）随机过程，特别是随机增长模型，与“双曲”几何图形;和（2）过滤和混沌动力系统产生的时间序列的推理。在第一个领域，主要目标是了解“双曲性”如何影响某些基本随机过程的行为，包括随机游走，分支随机游走，接触过程和渗流，并阐明在这些过程中发生的不同相变的性质。在第二个领域，目标是开发统计程序，用于分析由“混沌”动力学控制的系统产生的时间序列数据。特别是，从确定性的情况下，但混乱的，系统的数据被外部“噪声”损坏将进行研究。研究的目标是：（a）确定何时以及如何从嘈杂的时间序列中提取混沌“信号”;（B）根据原始时间序列数据，开发有关动力学定律的推理程序。随机增长过程是描述种群在时间和空间上增长和扩散的粗糙数学模型。了解这些过程演化的数学规律，也有助于我们理解流行病是如何发展的，有利的基因是如何在大群体中传播的，以及信息是如何通过大型通信网络传播的。“相变”的概念，一个系统的质量行为的急剧变化，由一个小的变化，它的参数沉淀，是特别重要的，因为理解这种过渡的性质可能会产生影响的干预策略的发展。

Lalley Research will be conducted in two distinct areas: (1) stochastic processes, especially stochastic growth models, on graphs with 'hyperbolic' geometries; and (2) filtering and inference for time series produced by chaotic dynamical systems. In the first area, the primary objective is an understanding of how 'hyperbolicity' influences the behavior of certain fundamental stochastic processes, including random walks, branching random walks, contact processes, and percolation, and to elucidate the nature of the different phase transitions that occur in such processes. In the second area, the goal is to develop statistical procedures for analyzing time series data produced by systems governed by 'chaotic' dynamics. In particular, situations where data from deterministic, but chaotic, systems is corrupted by external 'noise' will be studied. The goals of the research will be (a) to determine when, and how, chaotic 'signals' can be extracted from noisy time series; and (b) to develop procedures for inference about the governing dynamical laws based on raw time series data. Stochastic growth processes are crude mathematical models for the growth and spread of populations in time and space. Understanding of the mathematical laws governing the evolution of such processes may also contribute to our understanding of how epidemics develop, how favorable genes are disseminated in large populations, and how information makes its way through large communications networks. The notion of a 'phase transition', a drastic change in the qualitative behavior of a system precipitated by a small change in the parameters governing it, is especially important, because understanding the nature of such transitions may have ramifications for the development of intervention strategies.