Short probabilistic proof of the Brascamp-Lieb and Barthe theorems

We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young’s convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell’s stochastic proof of the Pr'ekopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.