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Hypercontractivity of Hamilton-Jacobi equations. (English) Zbl 1038.35020

Using the equivalence of logarithmic Sobolev inequalities and hypercontractivity of the associated heat semigroup proved by L. Gross [Am. J. Math. 97(1975), 1061–1083 (1976; Zbl 0318.46049)], the authors show that logarithmic Sobolev inequalities are similarly related to hypercontractivity of the solutions of Hamilton-Jacobi equations. This provides an explication of the connection of logarithmic Sobolev inequalities with transportation cost inequalities, which has been recently investigated by F. Otto and C. Villani [J. Funct. Anal. 173, 361–400 (2000; Zbl 0985.58019)]. In particular, the authors recover in this way transportation cost inequalities from Brunn-Minkowski inequalities for the exponential measure.

MSC:

35F20 Nonlinear first-order PDEs
47D07 Markov semigroups and applications to diffusion processes
70H20 Hamilton-Jacobi equations in mechanics
60E15 Inequalities; stochastic orderings
35K05 Heat equation
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