1,493 research outputs found
MEXIT: Maximal un-coupling times for stochastic processes
Classical coupling constructions arrange for copies of the \emph{same} Markov
process started at two \emph{different} initial states to become equal as soon
as possible. In this paper, we consider an alternative coupling framework in
which one seeks to arrange for two \emph{different} Markov (or other
stochastic) processes to remain equal for as long as possible, when started in
the \emph{same} state. We refer to this "un-coupling" or "maximal agreement"
construction as \emph{MEXIT}, standing for "maximal exit". After highlighting
the importance of un-coupling arguments in a few key statistical and
probabilistic settings, we develop an explicit \MEXIT construction for
stochastic processes in discrete time with countable state-space. This
construction is generalized to random processes on general state-space running
in continuous time, and then exemplified by discussion of \MEXIT for Brownian
motions with two different constant drifts.Comment: 28 page
Balance between quantum Markov semigroups
The concept of balance between two state preserving quantum Markov semigroups
on von Neumann algebras is introduced and studied as an extension of conditions
appearing in the theory of quantum detailed balance. This is partly motivated
by the theory of joinings. Balance is defined in terms of certain correlated
states (couplings), with entangled states as a specific case. Basic properties
of balance are derived and the connection to correspondences in the sense of
Connes is discussed. Some applications and possible applications, including to
non-equilibrium statistical mechanics, are briefly explored.Comment: v1: 40 pages. v2: Corrections and small additions made, 41 page
Localization for Linearly Edge Reinforced Random Walks
We prove that the linearly edge reinforced random walk (LRRW) on any graph
with bounded degrees is recurrent for sufficiently small initial weights. In
contrast, we show that for non-amenable graphs the LRRW is transient for
sufficiently large initial weights, thereby establishing a phase transition for
the LRRW on non-amenable graphs. While we rely on the description of the LRRW
as a mixture of Markov chains, the proof does not use the magic formula. We
also derive analogous results for the vertex reinforced jump process.Comment: 30 page
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