Hypercontractivity of Hamilton-Jacobi equations

Sergey G. Bobkov, Ivan Gentil, Michel Ledoux

Research output: Contribution to journalArticlepeer-review

168 Scopus citations

Abstract

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly to hypercontractivity of solutions of Hamilton-Jacobi equations. By the infimum-convolution description of the Hamilton-Jacobi solutions, this approach provides a clear view of the connection between logarithmic Sobolev inequalities and transportation cost inequalities investigated recently by F. Otto and C. Villani. In particular, we recover in this way transportation from Brunn-Minkowski inequalities and for the exponential measure.

Original languageEnglish (US)
Pages (from-to)669-696
Number of pages28
JournalJournal des Mathematiques Pures et Appliquees
Volume80
Issue number7
DOIs
StatePublished - Sep 2001

Keywords

  • Brunn-Minkowski inequality
  • Hamilton-Jacobi equation
  • Hypercontractivity
  • Infimum-convolution
  • Logarithmic Sobolev inequality

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