Remarks on Decay of Correlations and Witten Laplacians Brascamp–Lieb Inequalities and Semiclassical Limit

As it appears in recent articles by Helffer or Sjöstrand and Naddaf–Spencer, the analysis, in the context of the statistical mechanics, of measures of the type exp−Φ(x) dxis connected with the analysis of suitable Witten Laplacians on 1-forms. For illustrating this point of view, we present here remarks about the Brascamp–Lieb inequalities and its extensions and prove the decay of the correlation in some cases whenΦis weakly non convex.