Duke Mathematical Journal

Regularization under diffusion and anticoncentration of the information content

Ronen Eldan and James R. Lee

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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.

Article information

Duke Math. J., Volume 167, Number 5 (2018), 969-993.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G15: Gaussian processes 58J35: Heat and other parabolic equation methods

Gaussian space Ornstein–Uhlenbeck semigroup regularization hypercontractivity


Eldan, Ronen; Lee, James R. Regularization under diffusion and anticoncentration of the information content. Duke Math. J. 167 (2018), no. 5, 969--993. doi:10.1215/00127094-2017-0048. https://projecteuclid.org/euclid.dmj/1515747886

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