1,843 research outputs found
Merging for inhomogeneous finite Markov chains, part II: Nash and log-Sobolev inequalities
We study time-inhomogeneous Markov chains with finite state spaces using Nash
and logarithmic-Sobolev inequalities, and the notion of -stability. We
develop the basic theory of such functional inequalities in the
time-inhomogeneous context and provide illustrating examples.Comment: Published in at http://dx.doi.org/10.1214/10-AOP572 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Time inhomogeneous Markov chains with wave-like behavior
Starting from a given Markov kernel on a finite set and a bijection
of , we construct and study a time inhomogeneous Markov chain whose kernel
at time is obtained from by transport of . We show that this
construction leads to interesting examples, and we obtain quantitative results
for some of these examples.Comment: Published in at http://dx.doi.org/10.1214/09-AAP661 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Connected Lie groups and property RD
For a locally compact group, property RD gives a control on the convolution
norm of any compactly supported measure in terms of the -norm of its
density and the diameter of its support. We give a complete classification of
those Lie groups with property RD.Comment: 29 page
On the divine clockwork: the spectral gap for the correspondence limit of the Nelson diffusion generator for the atomic elliptic state
The correspondence limit of the atomic elliptic state in three dimensions is
discussed in terms of Nelson's stochastic mechanics. In previous work we have
shown that this approach leads to a limiting Nelson diffusion and here we
discuss in detail the invariant measure for this process and show that it is
concentrated on the Kepler ellipse in the plane z=0. We then show that the
limiting Nelson diffusion generator has a spectral gap; thereby proving that in
the infinite time limit the density for the limiting Nelson diffusion will
converge to its invariant measure. We also include a summary of the Cheeger and
Poincare inequalities both of which are used in our proof of the existence of
the spectral gap.Comment: 30 pages, 5 figures, submitted to J. Math. Phy
Single File Diffusion of particles with long ranged interactions: damping and finite size effects
We study the Single File Diffusion (SFD) of a cyclic chain of particles that
cannot cross each other, in a thermal bath, with long ranged interactions, and
arbitrary damping. We present simulations that exhibit new behaviors
specifically associated to systems of small number of particles and to small
damping. In order to understand those results, we present an original analysis
based on the decomposition of the particles motion in the normal modes of the
chain. Our model explains all dynamic regimes observed in our simulations, and
provides convincing estimates of the crossover times between those regimes.Comment: 30 pages, 9 figure
IL-1α and TNF-α Down-Regulate CRH Receptor-2 mRNA Expression in the Mouse Heart
Two receptors (CRH receptor type 1 and CRH receptor type 2) have been identified for the stress-induced neuropeptide, CRH and related peptides, urocortin, and urocortin II. We previously found marked down-regulation of cardiac CRH receptor type 2 expression following administration of bacterial endotoxin, lipopolysaccharide, a model of systemic immune activation, and inflammation. We postulated that inflammatory cytokines may regulate CRH receptor type 2. We show that systemic IL-1α administration significantly down-regulates CRH receptor type 2 mRNA in mouse heart. In addition, TNFα treatment also reduces CRH receptor type 2 mRNA expression, although the effect was not as marked as with IL-1α. However, CRH receptor type 2 mRNA expression is not altered in adult mouse ventricular cardiomyocytes stimulated in vitro with TNFα or IL-1α. Thus, cytokine regulation may be indirect. Exogenous administration of corticosterone in vivo or acute restraint stress also reduces cardiac CRH receptor type 2 mRNA expression, but like cytokines, in vitro corticosterone treatment does not modulate expression in cardiomyocytes. Interestingly, treatment with urocortin significantly decreases CRH receptor type 2 mRNA in cultured cardiomyocytes. We speculate that in vivo, inflammatory mediators such as lipopolysaccharide and/or cytokines may increase urocortin, which in turn down-regulates CRH receptor type 2 expression in the heart. Because CRH and urocortin increase cardiac contractility and coronary blood flow, impaired CRH receptor type 2 function during systemic inflammation may ultimately diminish the adaptive cardiac response to adverse conditions
Local regularity for parabolic nonlocal operators
Weak solutions to parabolic integro-differential operators of order are studied. Local a priori estimates of H\"older norms and
a weak Harnack inequality are proved. These results are robust with respect to
. In this sense, the presentation is an extension of Moser's
result in 1971.Comment: 31 pages, 3 figure
Spatial Mixing and Non-local Markov chains
We consider spin systems with nearest-neighbor interactions on an -vertex
-dimensional cube of the integer lattice graph . We study the
effects that exponential decay with distance of spin correlations, specifically
the strong spatial mixing condition (SSM), has on the rate of convergence to
equilibrium distribution of non-local Markov chains. We prove that SSM implies
mixing of a block dynamics whose steps can be implemented
efficiently. We then develop a methodology, consisting of several new
comparison inequalities concerning various block dynamics, that allow us to
extend this result to other non-local dynamics. As a first application of our
method we prove that, if SSM holds, then the relaxation time (i.e., the inverse
spectral gap) of general block dynamics is , where is the number of
blocks. A second application of our technology concerns the Swendsen-Wang
dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies
an bound for the relaxation time. As a by-product of this implication we
observe that the relaxation time of the Swendsen-Wang dynamics in square boxes
of is throughout the subcritical regime of the -state
Potts model, for all . We also prove that for monotone spin systems
SSM implies that the mixing time of systematic scan dynamics is . Systematic scan dynamics are widely employed in practice but have
proved hard to analyze. Our proofs use a variety of techniques for the analysis
of Markov chains including coupling, functional analysis and linear algebra
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