Research in Stochastic Processes and Nonlinear Filtering

Lalley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，在“双曲”几何图形上；(2)混沌动力系统产生的时间序列的滤波与推理。在第一个领域，主要目标是理解“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、接触过程和渗透，并阐明在这些过程中发生的不同相变的本质。在第二个领域，目标是开发用于分析由“混沌”动力学控制的系统产生的时间序列数据的统计程序。特别是，来自确定性但混乱的系统的数据被外部“噪声”破坏的情况将被研究。研究的目标将是(a)确定何时以及如何从噪声时间序列中提取混沌“信号”；(b)开发基于原始时间序列数据的控制动力学定律推断程序。随机增长过程是种群在时间和空间上增长和扩散的粗略数学模型。对控制这些过程进化的数学规律的了解，也可能有助于我们理解流行病是如何发展的，有利基因是如何在大量人群中传播的，以及信息是如何通过大型通信网络传播的。“相变”的概念尤其重要，因为理解这种转变的本质可能会对干预策略的发展产生影响，相变是由控制系统的参数的微小变化引起的系统定性行为的急剧变化。

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.