Generalized Logarithmic Sobolev Inequality by the JKO Scheme

TL;DR

This paper introduces a novel approach combining the Bakry-Émery method and optimal transport to establish generalized logarithmic Sobolev inequalities for log-concave measures, enhancing understanding of entropy and Fisher information dissipation.

Contribution

It develops a new discrete Bakry-Émery method using the JKO scheme to derive generalized inequalities under convexity assumptions, bridging existing techniques.

Findings

New generalized logarithmic Sobolev inequalities for log-concave measures.

Method recovers classical inequalities as special cases.

Provides a unified framework connecting Bakry-Émery and optimal transport methods.

Abstract

Using a discrete Bakry-{\'E}mery method based on the JKO scheme, relying on the dissipation of entropy and Fisher information along a discrete flow, we establish new generalized logarithmic Sobolev inequality for log-concave measures of the form $e^{-V}$ under strict convexity assumptions on $V$ . We then show how this method recovers some well-known inequalities. This approach can be viewed as interpolating between the Bakry-{\'E}mery method and optimal transport techniques based on geodesic convexity.