## Duke Mathematical Journal

### Regularization under diffusion and anticoncentration of the information content

#### Abstract

Under the Ornstein–Uhlenbeck semigroup $\{U_{t}\}$, any nonnegative measurable $f:\mathbb{R}^{n}\to\mathbb{R}_{+}$ exhibits a uniform tail bound better than that implied by Markov’s inequality and conservation of mass. For every $\alpha\geq e^{3}$, and $t\gt 0$,

$$\gamma_{n}(\{x\in\mathbb{R}^{n}:U_{t}f(x)\gt \alpha\int f\,d\gamma_{n}\})\leq C(t)\frac{1}{\alpha}\sqrt{\frac{\log\log\alpha}{\log\alpha}},$$ where $\gamma_{n}$ 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 $f:\mathbb{R}^{n}\to\mathbb{R}_{+}$ is semi-log-convex in the sense that for some $\beta\gt 0$, for all $x\in\mathbb{R}^{n}$, the eigenvalues of $\nabla^{2}\log f(x)$ are at least $-\beta$. Then $f$ satisfies a tail bound asymptotically better than that implied by Markov’s inequality.

#### Article information

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

Dates
Revised: 17 September 2017
First available in Project Euclid: 12 January 2018

https://projecteuclid.org/euclid.dmj/1515747886

Digital Object Identifier
doi:10.1215/00127094-2017-0048

Mathematical Reviews number (MathSciNet)
MR3782065

Zentralblatt MATH identifier
06870398

#### Citation

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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