# Science at the triple point between mathematics, mechanics and materials science

## Publication 115

### Quantitative Logarithmic Sobolev Inequalities and Stability Estimates

##### Authors:

M. Fathi
Université Pierre et Marie Curie
Paris, France

E. Indrei
Center for Nonlinear Analysis
Carnegie Mellon University
Pittsburgh PA 15213-3890 USA

M. Ledoux
University of Toulouse
Toulouse, France
and
Institut Universitaire de France

##### Abstract:
We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincaré inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an ${\rm L}^1$-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the context of the Bakry-Émery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.
##### Get the paper in its entirety
14-CNA-027.pdf

Back to Publications