2014/10/22 by Yuan Liu, Liu, Yuan
Mathematics · #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1410.6080
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-Guillin-Wang-Wu [8] to study functional inequalities, and the above result on LSI was first proved subject to ϕ(⋅)=d2(⋅, x0) by Cattiaux-Guillin-Wu [9] 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.