A link between the log-Sobolev inequality and Lyapunov condition

Abstract

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 $\varphi$-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 $\varphi (\cdot )=d^2(\cdot ,x_0)$ by Cattiaux-Guillin-Wu [9] through a combination of detective $L^2$ transportation-information inequality ${\mathrm{W}}_2\mathrm{I}$ 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.

8 pages, modified according to the referee's review, more minor corrections in version 4

References in corpus (1)

• Transportation cost-information inequalities and applications to random dynamical systems and diffusions