The optimal evolution of the free energy of interacting gases and its applications

Agueh, Martial; Ghoussoub, Nassif; Kang, Xiaosong

Comptes Rendus. Mathématique, Tome 337 (2003) no. 3, pp. 173-178.

Nous établissons une inégalité reliant l'énergie totale – interne, potentielle et interactive – de deux densités de probabilité, leur distance de Wasserstein, leurs barycentres ainsi que leur entropie relative généralisée. Cette inégalité implique plusieurs des inégalités géométriques classiques, ainsi qu'une correspondence remarquable entre les solutions de certaines équations quasilinéaires (ou semi-linéaires) et les solutions stationnaires d'équations du type Fokker–Planck. On établit aussi des inégalités HWBI – généralisant les inégalités HWI de Otto et Villani [J. Funct. Anal. 173 (2) (2000) 361–400] et de Carrillo et al. [Rev. Math. Iberoamericana (2003)], où le « B » refère au nouveau terme barycentrique – dont découlent plusieurs inégalités gaussiennes classiques.

We establish an inequality for the relative total – internal, potential and interactive – energy of two arbitrary probability densities, their Wasserstein distance, their barycenters and their generalized relative Fisher information. This inequality leads to many known and powerful geometric inequalities, as well as to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker–Planck type equations. It also yields the HWBI inequalities – which extend the HWI inequalities in Otto and Villani [J. Funct. Anal. 173 (2) (2000) 361–400], and in Carrillo et al. [Rev. Math. Iberoamericana (2003)], with the additional ‘B’ referring to the new barycentric term – from which most known Gaussian inequalities can be derived.

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