Coupling and unimodularity in stationary settings

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.