Short probabilistic proof of the Brascamp-Lieb and Barthe theorems (Q2925377)

The author presents a new proof of the Brascamp-Lieb theorem (see [\textit{H. J. Brascamp} and \textit{E. H. Lieb}, Adv. Math. 20, 151--173 (1976; Zbl 0339.26020)] and [\textit{E. H. Lieb}, Invent. Math. 102, No. 1, 179--208 (1990; Zbl 0726.42005)]) and of the Barthe theorem (see [\textit{F. Barthe}, Invent. Math. 134, No. 2, 335--361 (1998; Zbl 0901.26010)]) from a probabilistic point of view. To this end, he uses a representation formula due to \textit{M. Boué} and \textit{P. Dupuis} [Ann. Probab. 26, No. 4, 1641--1659 (1998; Zbl 0936.60059)] for the quantity \(\ln\left(\int e^{g(x)}\gamma (dx)\right)\), where \(g:\mathbb{R}^n\rightarrow \mathbb{R}\) is a measurable map bounded from below, and \(\gamma\) is a Gaussian measure. This formula was rediscovered by \textit{C. Borell} [Potential Anal. 12, No. 1, 49--71 (2000; Zbl 0976.60065)] as a useful tool to prove some functional inequalities.NEWLINENEWLINEIn Sections 2 and 3, the theorems are proved under the additional hypothesis of the existence of a positive definite matrix \(A\) holding \(A^{-1}=\sum_{j=1}^{m} c_j B_j^{*}(B_j A B_j^{*})^{-1} B_j\) (here, \((c_1,B_1), \dots, (c_m,B_m)\) are the elements of a Brascamp-Lieb datum, with \(c_j>0\) and \(B_j:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n_j}\) linear and onto); and in Section 4, the author explains why finally the above results yield the Brascamp-Lieb and Barthe theorems.