Comment on: “Hypercontractivity of Hamilton–Jacobi equations”, by S. Bobkov, I. Gentil and M. Ledoux

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In their remarkable work [1], Bobkov, Gentil and Ledoux improve, generalize and simplify most previously known results about hypercontractivity, logarithmic Sobolev and transport inequalities. Among other results, they are able to recover and generalize the HWI inequalities introduced in [4], (1) DV ≥ KId ⇒ H(f |e−V ) ≤ W (f, e−V ) √ I(f |e−V )− K 2 W (f, e−V ). Here f and g are arbitrary probability distribution functions, H(f |g) ≡ ∫ f log(f/g) is the Kullback relative entropy of f w.r.t g, I(f |g) ≡ ∫ f |∇ log(f/g)|2 is the relative Fisher information of f w.r.t g, and