Advanced Search

We prove a stochastic formula for the Gaussian relative entropy in the spirit of Borell’s formula for the Laplace transform. As an application, we give simple proofs of a number of functional inequalities.

We give a proof, based on the Poincaré inequality, of the symmetric property ($\tau$) for the Gaussian measure. If $f:{ℝ}^d\to ℝ$ is continuous, bounded from below and even, we define $Hf\left(x\right)={inf}_{y}f(x+y)+\frac{1}{2}{\left|y\right|}^2$ and we have $$\int {\mathrm{e}}^{-f}d{\gamma }_d\int {\mathrm{e}}^{Hf}d{\gamma }_d\le 1.$$ This property is equivalent to a certain functional form of the Blaschke-Santaló inequality, as explained in a paper by Artstein, Klartag and Milman.