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Remarks on Gaussian Noise Stability, Brascamp-Lieb and Slepian Inequalities

Part of the Lecture Notes in Mathematics book series (LNM,volume 2116)

Abstract

E. Mossel and J. Neeman recently provided a heat flow monotonicity proof of Borell’s noise stability theorem. In this note, we develop the argument to include in a common framework noise stability, Brascamp-Lieb inequalities (including hypercontractivity), and even a weak form of Slepian inequalities. The scheme applies furthermore to families of measures with are more log-concave than the Gaussian measure.

Keywords

  • Gaussian Measure
  • Logarithmic Sobolev Inequality
  • Gaussian Vector
  • Noise Stability
  • Multidimensional Version

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Acknowledgements

This note grew up out of discussions and exchanges with J. Neeman and E. Mossel around their works [28, 30] and [18]. In particular, Sect. 2, the first part of Sects. 3 and 5 are directly following their contributions. I sincerely thank them for their interest and comments. I also thank K. Oleszkiewicz for exchanges on ρ-concave functions, R. Bouyrie for several comments and corrections, F. Barthe for pointing out the reference [17], and the referee for helpful comments in improving the exposition.

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Ledoux, M. (2014). Remarks on Gaussian Noise Stability, Brascamp-Lieb and Slepian Inequalities. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_20

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