Regularization under diffusion and anti-concentration of temperature

Consider a non-negative function f : R → R+ such that ∫ f dγn = 1, where γn is the ndimensional Gaussian measure. If f is semi-log-convex, i.e. if there exists a number β ≥ 1 such that for all x ∈ R, the eigenvalues of ∇ log f(x) are at least −β, then f satisfies an improved form of Markov’s inequality: For all α ≥ e, γn ( {x ∈ R : f(x) > α} ) ≤ 1 α · Cβ(log logα) 4 √ logα , where C is a universal constant. The bound is optimal up to a factor of C √ β(log logα), 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 [Tal89].