Doeblin Trees

This paper is centered on the random graph generated by a Doeblin-type coupling of discrete time processes on a countable state space 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 at any time. The bridge graph is made into a unimodular network by marking it and selecting a root in a specified fashion. The unimodularity of this network is leveraged to discern global properties of the larger Doeblin graph. Bi-recurrence, i.e., recurrence both forwards and backwards in time, is introduced and shown to be a key property in uniquely distinguishing paths in the Doeblin graph, and also a decisive property for Markov chains indexed by $\mathbb{Z}$. Properties related to simulating the bridge graph are also studied.Comment: 44 pages, 4 figure

Processes on Unimodular Random Networks

We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications to stochastic comparison of continuous-time random walk.Comment: 66 pages; 3rd version corrects formula (4.4) -- the published version is incorrect --, as well as a minor error in the proof of Proposition 4.10; 4th version corrects proof of Proposition 7.1; 5th version corrects proof of Theorem 5.1; 6th version makes a few more minor correction

Unimodularity and preservation of volumes in nonholonomic mechanics

The equations of motion of a mechanical system subjected to nonholonomic linear constraints can be formulated in terms of a linear almost Poisson structure in a vector bundle. We study the existence of invariant measures for the system in terms of the unimodularity of this structure. In the presence of symmetries, our approach allows us to give necessary and sufficient conditions for the existence of an invariant volume, that unify and improve results existing in the literature. We present an algorithm to study the existence of a smooth invariant volume for nonholonomic mechanical systems with symmetry and we apply it to several concrete mechanical examples.Comment: 37 pages, 3 figures; v3 includes several changes to v2 that were done in accordance to the referee suggestion

A Characterization of Norm Compactness in the Bochner Space Lp(G;B)L^p (G ; B) For an Arbitrary Locally Compact Group GG

In this paper, we generalize a result of N. Dinculeanu which characterizes norm compactness in the Bochner space $L^p(G ; B)$ in terms of an approximate identity and translation operators, where $G$ is a locally compact abelian group and $B$ is a Banach space. Our characterization includes the case where $G$ is nonabelian, and we weaken the hypotheses on the approximate identity used, providing new results even for the case $B = \mathbb{C}$ and $G = \mathbb{R}^n.$Comment: 13 pages, accepted into the "Journal of Mathematical Analysis and Applications.