1,322 research outputs found
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 . 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 For an Arbitrary Locally Compact Group
In this paper, we generalize a result of N. Dinculeanu which characterizes
norm compactness in the Bochner space in terms of an approximate
identity and translation operators, where is a locally compact abelian
group and is a Banach space. Our characterization includes the case where
is nonabelian, and we weaken the hypotheses on the approximate identity
used, providing new results even for the case and Comment: 13 pages, accepted into the "Journal of Mathematical Analysis and
Applications.
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