Opérateur d’inf-convolution et équations d’Hamilton-Jacobi sur les graphes

This work introduces a notion of gradient and an infimum-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity of this class of infimal-convolution operators is connected to some discrete version of the log-Sobolev inequality and to a discrete version of Talagrand's transport inequality.