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 $\phi$-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 $\phi(\cdot)=d^2(\cdot, x_0)$ by Cattiaux-Guillin-Wu [9] through a combination of detective $L^2$ transportation-information inequality $\mathrm{W_2I}$ 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 $\mu e^{\delta d^2(x,x_0)} 0$ 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