Research in Stochastic Processes and Nonlinear Filtering

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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.