130,698 research outputs found
A link between the log-Sobolev inequality and Lyapunov condition
We give an alternative look at the log-Sobolev inequality (LSI in short) for
log-concave measures by semigroup tools. The similar idea yields a heat flow
proof of LSI under some quadratic Lyapunov condition for symmetric diffusions
on Riemannian manifolds provided the Bakry-Emery's curvature is bounded from
below. Let's mention that, the general -Lyapunov conditions were
introduced by Cattiaux-Guillin-Wang-Wu [8] to study functional inequalities,
and the above result on LSI was first proved subject to by Cattiaux-Guillin-Wu [9] through a combination of detective
transportation-information inequality and the HWI inequality of
Otto-Villani.
Next, we assert a converse implication that the Lyapunov condition can be
derived from LSI, which means their equivalence in the above setting.Comment: 8 pages, modified according to the referee's review, more minor
corrections in version
Gaussian integrability of distance function under the Lyapunov condition
In this note we give a direct proof of the Gaussian integrability of distance
function as provided
the Lyapunov condition holds for symmetric diffusion Markov operators, which
answers a question proposed in Cattiaux-Guillin-Wu [6, Page 295]. The similar
argument still works for diffusions processes with unbounded diffusion
coefficients and for jump processes such as birth-death chains. An analogous
discussion is also made under the Gozlan's condition arising from [9,
Proposition 3.5].Comment: 11 pages, published, ECP. Some extensions to unbounded diffusions and
jump processes have been added, and two referees' suggestions incorporate
Finding binary active galactic nuclei by the centroid shift in imaging surveys
The census of binary active galactic nuclei (AGNs) is important in order to
understand the merging history of galaxies and the triggering of AGNs. However,
there is still no efficient method for selecting the candidates of binary AGNs.
The non-synchronous variations of the two AGNs in one binary system will induce
the shift of the image centroid. Since the astrometric error is normally much
smaller than the angular resolution of telescopes, it is possible to detect
such shifts even in the unresolved system via multi-epoch observations. We
perform some simulations and find that hundreds of observations are required to
discover compact binary AGNs. This method is suitable for the future
large-scale surveys, e.g., the Large Synoptic Survey Telescope, and it might
lead to a large sample of binary AGNs with a 1-2 yr survey.Comment: 4 pages, 4 figures, published in Astronomy & Astrophysic
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