Abstract
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 generalised 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 self-adjointness and domain characterisations.
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Supported by TMR grant FMRX-CT960001 of the European Commision, ‘PDE and QM’ at Université de Paris-Sud, France; partly by the Danish Natural Sciences Research Council, grant 9700987.
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Johnsen, J. On the spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb's inequality. Integr equ oper theory 36, 288–324 (2000). https://doi.org/10.1007/BF01213926
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DOI: https://doi.org/10.1007/BF01213926