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 …