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 ϕ-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 ϕ(⋅) = d 2(⋅,x 0) by Cattiaux et al. (Proba. Theory Relat. Fields 148(1–2), 285–304 2010) through a combination of detective L 2 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.
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Liu, Y. A Link Between the Log-Sobolev Inequality and Lyapunov Condition. Potential Anal 44, 629–637 (2016). https://doi.org/10.1007/s11118-015-9522-1
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DOI: https://doi.org/10.1007/s11118-015-9522-1