A note on Talagrand's transportation inequality and logarithmic Sobolev inequality
Patrick Cattiaux (LSProba), Arnaud Guillin, Liming Wu
TL;DR
This paper presents simple Lyapunov-based conditions that ensure Talagrand's transportation inequality and the logarithmic Sobolev inequality, applicable even without lower bounds on Bakry-Emery curvature, along with new examples.
Contribution
It introduces straightforward Lyapunov conditions for key inequalities, expanding applicability beyond curvature bounds and providing new illustrative examples.
Findings
Lyapunov conditions suffice for Talagrand's and logarithmic Sobolev inequalities
Conditions apply without Bakry-Emery curvature lower bounds
Several new examples demonstrate the results
Abstract
We give by simple arguments sufficient conditions, so called Lyapunov conditions, for Talagrand's transportation information inequality and for the logarithmic Sobolev inequality. Those sufficient conditions work even in the case where the Bakry-Emery curvature is not lower bounded. Several new examples are provided.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Geometry and complex manifolds