Singular perturbations of weakly coupled systems of Hamilton Jacobi equations

In this paper we study singular perturbations of weakly coupled systems of elliptic equations. A model problem is given by small random perturbations of random evolution processes. In this setting we give a PDE proof of large deviation results, analogous to those studied in Ann. Probab. 15 (1987), 646-658, Stochastics Stochastics Rep. 33 (1990), 111-148, Probab. Theory Related Fields 94 (1993), 335-374. An essential tool in our approach is the weak KAM theory introduced in Calc. Var. Partial Differential Equations 22 (2005), 185-228.