On the spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb's inequality

A study is made of an integral identity of Helffer and Sj{\"o}strand, which for some class of probability measures yields a formula for the covariance of two functions (of a stochastic variable). In comparison with the Brascamp--Lieb inequality, this formula is a more flexible and in some contexts stronger means for the analysis of correlation asymptotics in statistical mechanics. Using a fine version of the Closed Range Theorem, the identity's validity is shown to be equivalent to some explicitly given spectral properties of Witten-Laplacians on Euclidean space, and the formula is moreover deduced from the obtained abstract expression for the range projection. As a corollary, a generalised version of Brascamp--Lieb's inequality is obtained. For a certain class of measures occurring in statistical mechanics, explicit criteria for the means are found from the Persson--Agmon formula, from compactness of embeddings and from the Weyl calculus of pseudo-differential operators, which give results for closed range, strict positivity, essential self-adjointness and domain characterisations.