A SHORT PROOF OF THE “CONCAVITY OF ENTROPY POWER”

A SHORT PROOF OF THE “CONCAVITY OF ENTROPY POWER” C. VILLANI Abstract. We give a simple proof of the “concavity of entropy power”. Key words : Entropy power, Fisher information, heat semigroup. Let f be a probability measure on Rn. We define the action of the heat semigroup (Pt)t≥0 on f , by the solution of the partial differential equation ∂ ∂tPtf = ∆(Ptf). Equivalently, Ptf is the convolution of f with the n-dimensional Gauss- ian density having mean vector 0 and covariance matrix 2tIn, where In is the identity matrix. The “concavity of entropy power” theorem states that (1) d 2 dt2N(Ptf) ≤ 0. Here N(f) = e 2H(f) n 2pie , H(f) = ? ∫ R n f log f. The functional N(f) is the so-called ”entropy power” of f , as intro- duced by Shannon, while H(f) is Shannon's entropy functional (which coincides to Boltzmann's entropy up to a change of sign). The normal- izing factor 2pie is nonessential and we mention it only to stick to the conventions of Shannon. Inequality (1) is due to Costa [4].

• entropy power


• dimensional gauss- ian


• proof does


• ecole normale


• ?u ·


• blachman-stam inequality


• boltzmann's entropy up


• remainder term


• differential equation