The strong data processing inequality under the heat flow

Let ν and μ be probability distributions on Rⁿ, and ν_s,μ_s 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(ν_s|μ_s) for various divergences, including the χ² and Kullback-Leibler (KL) divergences. We prove upper and lower bounds on the strong dataprocessing 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 ν_s. As a byproduct of our analysis, we obtain new lower bounds on the mutual information between X and Y = X + √sZ, 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 Poincaré constant of X.