Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp.Lieb inequalities

In this work, we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp.Lieb inequalities. For this, we use optimal transport methods and the Borell.Brascamp.Lieb inequality. These refinements can be written as a deficit in the classical inequalities. They have the right scale with respect to the dimension. They lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behavior for the laws of solutions to stochastic differential equations.