  Following the random approach of Mitake, Siconolfi,Tran and Yamada, we define
a Lax--Oleinik formula adapted to evolutive weakly coupled systems of
Hamilton--Jacobi equations. It is reminiscent of the corresponding scalar
formula, with the relevant difference that it has a stochastic character since
it involves, loosely speaking, random switchings between the various associated
Lagrangians. We prove that the related value functions are viscosity solutions
to the system, and establish existence of minimal random curves under fairly
general hypotheses. Adding Tonelli like assumptions on the Hamiltonians, we
show differentiability properties of such minimizers, and existence of adjoint
random curves. Minimizers and adjoint curves are trajectories of a twisted
generalized Hamiltonian dynamics.
