Coupling and unimodularity in stationary settings
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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.