Duke Mathematical Journal
- Duke Math. J.
- Volume 167, Number 5 (2018), 969-993.
Regularization under diffusion and anticoncentration of the information content
Under the Ornstein–Uhlenbeck semigroup , any nonnegative measurable exhibits a uniform tail bound better than that implied by Markov’s inequality and conservation of mass. For every , and ,
where is the -dimensional Gaussian measure and is a constant depending only on . 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 is semi-log-convex in the sense that for some , for all , the eigenvalues of are at least . Then satisfies a tail bound asymptotically better than that implied by Markov’s inequality.
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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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