Singular perturbations of weakly coupled systems of Hamilton–Jacobi equations

Abstract

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.