Representation formula for the entropy and functional inequalities. (English. French summary) Zbl 1279.39011
The purpose of the present paper is to prove a representation formula for the Gaussian relative entropy, both in \(\mathbb{R}^d\) and in the Wiener space, providing the entropy counterpats of C. Borell’s formula for the Laplace transform. All these formulas have a common feature: Girsanov’s theorem. However, the author’s approach is somewhat different from that of C. Borell [Potential Anal. 12, No. 1, 49–71 (2000; Zbl 0976.60065)] and M. Boué and P. Dupuis [Ann. Probab. 26, No. 4, 1641–1659 (1998; Zbl 0936.60059)]: it draws a connection with the work of H. Föllmer [Time reversal on Wiener space. Stochastic processes – mathematics and physics, Proc. 1st BiBoS-Symp., Bielefeld/Ger. 1984, Lect. Notes Math. 1158, 119–129 (1986; Zbl 0582.60078) and Random fields and diffusion processes. Calcul des probabilités, Éc. d’Été, Saint-Flour/Fr. 1985–87, Lect. Notes Math. 1362, 101–203 (1988; Zbl 0661.60063)] which makes the whole argument more simple. As an application, unified new proofs of a number of Gaussian inequalities (such as: the tranportation cost inequality; logarithmic Sobolev inequality; Shannon’s inequality; Brascamp-Lieb inequality; reversed Brascamp-Lieb inequality) are offered.
Reviewer: József Sándor (Cluj-Napoca)
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