A link between the log-Sobolev inequality and Lyapunov condition

TL;DR

This paper explores the connection between the log-Sobolev inequality and Lyapunov conditions for symmetric diffusions, providing new proofs and establishing their equivalence under certain geometric and functional conditions.

Contribution

It offers a heat flow proof of LSI using Lyapunov conditions and demonstrates the equivalence between LSI and Lyapunov conditions in this context.

Findings

Heat flow proof of LSI under quadratic Lyapunov condition

Equivalence between LSI and Lyapunov conditions for symmetric diffusions

Extension of previous results to more general Lyapunov functions

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 $\phi$-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.