[Paper Review] Mass transport and variants of the logarithmic Sobolev inequality

Franck Barthe, Alexander V. Kolesnikov|arXiv (Cornell University)|Sep 25, 2007

Nonlinear Partial Differential Equations38 references4 citations

TL;DR

This paper establishes new sufficient conditions for modified logarithmic Sobolev inequalities and isoperimetric inequalities using optimal transport methods, particularly focusing on measures with non-convex potentials that still satisfy functional inequalities due to strong integrability. The key contribution is a unified framework that generalizes known criteria and proves that Riemannian measures with lower Ricci curvature bounds and specific decaying potentials satisfy modified log-Sobolev and $F$-inequalities for $1 < \alpha \leq 2$, extending results to non-Gaussian and non-log-concave measures.

ABSTRACT

We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting.

Motivation & Objective

• To develop a systematic optimal transport approach to modified logarithmic Sobolev and isoperimetric inequalities.
• To identify sufficient conditions under which measures with non-convex potentials still satisfy functional inequalities, relying on strong integrability to counteract lack of convexity.
• To generalize known criteria for logarithmic Sobolev inequalities to the Riemannian setting and unify existing results via transportation techniques.
• To establish new functional inequalities—specifically $I(\tau)$, $F_\tau$, and modified log-Sobolev—for measures with power-type decay $e^{-|x|^\alpha}$, $1 < \alpha \leq 2$, on manifolds with Ricci curvature bounded below.

Proposed method

• Utilizes optimal transport theory to derive functional inequalities from isoperimetric properties, particularly through the use of cost functions and their convex conjugates.
• Applies 'tightening' techniques to reduce general inequalities to simpler forms, enabling the use of known inequalities on perturbed measures.
• Employs integration by parts and the Calabi lemma to handle non-smooth potentials (e.g., distance functions) on Riemannian manifolds, especially around cut loci.
• Uses perturbation arguments and boundedness of derivatives of logarithmic functions to control error terms in gradient estimates.
• Applies the Euclidean logarithmic Sobolev inequality as a base, then extends it to weighted Riemannian measures via a change of measure and perturbation of the potential.
• Relies on the bound $\Delta p \leq C_1 + C_2 r^{\alpha-1}$ and $|\nabla p|^2 \sim r^{2(\alpha-1)}$ to dominate curvature effects and apply Cauchy-Schwarz and Young’s inequality in the final estimates.

Experimental results

Research questions

• RQ1 Under what conditions does a measure with a non-convex potential still satisfy a modified logarithmic Sobolev inequality, despite lacking strong convexity?
• RQ2 Can optimal transport methods unify and generalize existing criteria for logarithmic Sobolev and $F$-Sobolev inequalities in both Euclidean and Riemannian settings?
• RQ3 How do isoperimetric inequalities relate to modified log-Sobolev and $F$-Sobolev inequalities in the context of measures with power-type tails?
• RQ4 What is the role of integrability properties in balancing the lack of convexity in the potential for functional inequalities to hold?
• RQ5 Can the Euclidean logarithmic Sobolev inequality be extended to Riemannian manifolds with lower Ricci curvature bounds and non-Gaussian measures of the form $e^{-N\rho^\alpha}$?

Key findings

• The paper proves that for $1 < \alpha \leq 2$, the measure $\mu = \frac{1}{Z_{\alpha,N}} \exp(-N\rho^\alpha(x,x_0))\,dv$ on a Riemannian manifold with $\mathrm{Ric} \geq K\,\mathrm{Id}$ satisfies the $I(\tau)$-inequality with $\tau = 2(1 - 1/\alpha)$.
• Such measures also satisfy the $F_\tau$-inequality and a modified log-Sobolev inequality with cost function $c_{\alpha}$, which implies improved concentration and semigroup properties.
• The proof relies on transforming the Euclidean log-Sobolev inequality into a weighted form via a change of measure and applying perturbation techniques to handle non-smooth potentials.
• The method shows that even when the potential is not convex, strong integrability (e.g., $e^{\varepsilon r^\alpha} \in L^1(\nu)$) allows the functional inequalities to hold.
• The result generalizes previous findings for $\alpha \geq 2$ to the critical range $1 < \alpha < 2$, where the potential is less regular but still allows for functional inequalities.
• The framework unifies known criteria for $F$-Sobolev and super-Poincaré inequalities and recovers them via transportation methods, including Wang’s correspondence between $F$-inequalities and super-Poincaré inequalities.