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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.

MSC:
58J05Elliptic equations on manifolds, general theory
82B05Classical equilibrium statistical mechanics (general)
47A05General theory of linear operators
47A07Forms on topological linear spaces
58J10Differential complexes; elliptic complexes
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