The strong data processing inequality under the heat flow

Let ν and μ be probability distributions on Rⁿ, and νₛ,μₛ be their evolution under the heat flow, that is, the probability distributions resulting from convolving their density with the density of an isotropic Gaussian random vector with variance s in each entry. This paper studies the rate of decay of s↦ D(νₛ\|μₛ) for various divergences, including the χ² and Kullback-Leibler (KL) divergences. We prove upper and lower bounds on the strong data-processing inequality (SDPI) coefficients corresponding to the source μ and the Gaussian channel. We also prove generalizations of de Brujin's identity, and Costa's result on the concavity in s of the differential entropy of νₛ. As a byproduct of our analysis, we obtain new lower bounds on the mutual information between X and Y=X+√s Z, where Z is a standard Gaussian vector in Rⁿ, independent of X, and on the minimum mean-square error (MMSE) in estimating X from Y, in terms of the Poincar\'e constant of X.