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    Coupling and unimodularity in stationary settings

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    MURPHY-DISSERTATION-2019.pdf (1.966Mb)
    Author
    Murphy, James Thomas, III
    0000-0002-5530-8446
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    Abstract
    This dissertation studies three applications of the tools of coupling and unimodularity in stationary settings. The first application is to exact coupling of random walks. Conditions for admitting a successful exact coupling are given that are necessary and in the Abelian case also sufficient. This solves a problem posed by H. Thorisson. The second application is centered on the random graph generated by a Doeblin-type coupling of discrete time processes whereby when two paths meet, they merge. This random graph is studied through a novel subgraph, called a bridge graph, generated by paths started in a fixed state. The bridge graph is then made into a unimodular network. The final application focuses on point-shifts of point processes on topological groups. Foliations and connected components generated by point-shifts are studied, and the cardinality classification of connected components is generalized to unimodular groups.
    Department
    Mathematics
    Subject
    Exact coupling
    Random walk
    Doeblin tree
    Eternal family tree
    Unimodular network
    Mass-transport
    Point-shift
    Point process
    URI
    https://hdl.handle.net/2152/75807
    http://dx.doi.org/10.26153/tsw/2909
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    © The University of Texas at Austin