Hypercontractivity of Hamilton-Jacobi equations

Following the equivalence between logarithmic Sobolev inequalities and hype rcontractivity showed by L. Gross, we prove that logarithmic Sobolev inequa lities are related similarly to hypercontractivity of solutions of Hamilton -Jacobi equations. By the infimum-convolution description of the Hamilton-J acobi solutions, this approach provides a clear view of the connection betw een logarithmic Sobolev inequalities and transportation cost inequalities i nvestigated recently by F. Otto and C. Villani. In particular, we recover i n this way transportation from Brunn-Minkowski inequalities and for the exp onential measure. (C) 2001 Editions scientifiques et medicales Elsevier SAS .