Abstract
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.
Citation
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
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