Regularization under diffusion and anticoncentration of the information content

Abstract

Under the Ornstein–Uhlenbeck semigroup $\left\{U_t\right\}$, 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 $\alpha \ge e^3$, and $t>0$,

$${\gamma }_n\left(\right\{x\in {\mathbb{R}}^n:U_{t}f\left(x\right)>\alpha \int fd{\gamma }_n\left\}\right)\le C\left(t\right)\frac{1}{\alpha }\sqrt{\frac{loglog\alpha }{log\alpha }},$$ where ${\gamma }_n$ is the $n$-dimensional Gaussian measure and $C\left(t\right)$ 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 $\beta >0$, for all $x\in {\mathbb{R}}^n$, the eigenvalues of ${\nabla }^{2}logf\left(x\right)$ are at least $-\beta$. Then $f$ satisfies a tail bound asymptotically better than that implied by Markov’s inequality.