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 et al. (J. Funct. Anal. 256(6), 1821–1841 2009) to study functional inequalities, and the above result on LSI was first proved subject to ?(?) = d²(?,x₀) by Cattiaux et al. (Proba. Theory Relat. Fields 148(1–2), 285–304 2010) through a combination of detective L² transportation-information inequality W₂I 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.