Abstract

The aim of this paper is to study the behavior of solutions of some nonlinear partial differential equations of Mac Kean-Vlasov type. The main tools used are, on one hand, the logarithmic Sobolev inequality and its connections with the concentration of measure and the transportation inequality with quadratic cost; on the other hand, the propagation of chaos for particle systems in mean field interaction. Key words: Interacting particle system, logarithmic Sobolev inequality, propagation of chaos, relative entropy, concentration of measure 1 Introduction A probability measure on R n satisfies a logarithmic Sobolev inequality with constant C if Ent i f 2 j C Z jrf j 2 d (1) for all smooth enough functions f where Ent i f 2 j = Z f 2 log f 2 d \Gamma Z f 2 d log Z f 2 d : Let us recall two consequences of this property for . First, for every r 0, and every Lipschitz function f on R n (equiped with the Euclidean topology) with kfk Lip = sup x6=y jf(...

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