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On the spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb’s inequality. (English) Zbl 1023.58012
Summary: A study is made of a recent integral identity of B. Helffer and J. Sjöstrand, which for a not yet fully determined 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 generalized version of Brascamp-Lieb’s inequality is obtained. For a certain class of measures occuring in statistical mechanics, explicit criteria for the Witten-Laplacians are found from the Persson-Agmon formula, from compactness of embeddings and from the Weyl calculus, which give results for closed range, strict positivity, essential selfadjointness and domain characterizations.

58J05 Elliptic equations on manifolds, general theory
82B05 Classical equilibrium statistical mechanics (general)
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A07 Forms (bilinear, sesquilinear, multilinear)
58J10 Differential complexes
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