The Lax-Oleinik semigroup on graphs

We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which happen to be the fixed points of the Lax-Oleinik semi-group, and identify their uniqueness set as the Aubry set, giving a representation formula. Our main result is the long time convergence of the Lax Oleinik semi-group. It follows that weak KAM solutions are viscosity solutions of the Hamilton-Jacobi equation [ 3 , 4 ], and in the case of Hamiltonians called of eikonal type in [ 3 ], we prove that the converse holds.