Regularization under diffusion and anti-concentration of temperature

Ronen Eldan, James R. Lee

Published 2014-10-14Version 1

Consider a non-negative function $f : \mathbb R^n \to \mathbb R_+$ such that $\int f\,d\gamma_n = 1$, where $\gamma_n$ is the $n$-dimensional Gaussian measure. If $f$ is semi-log-convex, i.e. if there exists a number $\beta \geq 1$ such that for all $x \in \mathbb R^n$, the eigenvalues of $\nabla^2 \log f(x)$ are at least $-\beta$, then $f$ satisfies an improved form of Markov's inequality: For all $\alpha > e^3$, \[ \gamma_n(\{x \in \mathbb R^n : f(x) > \alpha \}) \leq \frac{1}{\alpha} \cdot \frac{C\beta (\log \log \alpha)^{4}}{\sqrt{\log \alpha}}, \] where $C$ is a universal constant. The bound is optimal up to a factor of $C \sqrt{\beta} (\log \log \alpha)^{4}$, as it is met by translations and scalings of the standard Gaussian density. In particular, this implies that the mass on level sets of a probability density decays uniformly under the Ornstein-Uhlenbeck semigroup. This confirms positively the Gaussian case of Talagrand's convolution conjecture (1989).