  We study a class of weakly coupled Hamilton-Jacobi systems with a specific
aim to perform a qualitative analysis in the spirit of weak KAM theory. Our
main achievement is the definition of a family of related action functionals
containing the Lagrangians obtained by duality from the Hamiltonians of the
system. We use them to characterize, by means of a suitable estimate, all the
subsolutions of the system, and to explicitly represent some subsolutions
enjoying an additional maximality property. A crucial step for our analysis is
to put the problem in a suitable random frame. Only some basic knowledge of
measure theory is required, and the presentation is accessible to readers
without background in probability.
