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1 April 2018 Regularization under diffusion and anticoncentration of the information content
Ronen Eldan, James R. Lee
Duke Math. J. 167(5): 969-993 (1 April 2018). DOI: 10.1215/00127094-2017-0048

Abstract

Under the Ornstein–Uhlenbeck semigroup {Ut}, any nonnegative measurable f:RnR+ exhibits a uniform tail bound better than that implied by Markov’s inequality and conservation of mass. For every αe3, and t>0,

γn({xRn:Utf(x)>αfdγn})C(t)1αloglogαlogα, where γn is the n-dimensional Gaussian measure and C(t) is a constant depending only on t. This confirms positively the Gaussian limiting case of Talagrand’s convolution conjecture (1989). This is shown to follow from a more general phenomenon. Suppose that f:RnR+ is semi-log-convex in the sense that for some β>0, for all xRn, the eigenvalues of 2logf(x) are at least β. Then f satisfies a tail bound asymptotically better than that implied by Markov’s inequality.

Citation

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Ronen Eldan. James R. Lee. "Regularization under diffusion and anticoncentration of the information content." Duke Math. J. 167 (5) 969 - 993, 1 April 2018. https://doi.org/10.1215/00127094-2017-0048

Information

Received: 11 February 2016; Revised: 17 September 2017; Published: 1 April 2018
First available in Project Euclid: 12 January 2018

zbMATH: 06870398
MathSciNet: MR3782065
Digital Object Identifier: 10.1215/00127094-2017-0048

Subjects:
Primary: 60D05
Secondary: 58J35 , 60G15

Keywords: Gaussian space , hypercontractivity , Ornstein–Uhlenbeck semigroup , regularization

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 5 • 1 April 2018
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