A Lagrangian Approach to Weakly Coupled Hamilton--Jacobi Systems

Abstract

We perform a qualitative analysis of a class of weakly coupled Hamilton--Jacobi systems in the spirit of weak KAM theory. We define a family of related action functionals containing the Lagrangians associated with the Hamiltonians of the system. We use them to characterize the subsolutions of the system and to provide explicit representation formulae for subsolutions enjoying an additional maximality property. A crucial step for our analysis is to put the problem in a suitable random frame. The presentation is accessible to readers without a background in probability; only some basic knowledge of measure theory is required.