  We investigate the large-time behavior of the value functions of the optimal
control problems on the $n$-dimensional torus which appear in the dynamic
programming for the system whose states are governed by random changes. From
the point of view of the study on partial differential equations, it is
equivalent to consider viscosity solutions of quasi-monotone weakly coupled
systems of Hamilton--Jacobi equations. The large-time behavior of viscosity
solutions of this problem has been recently studied by the authors and Camilli,
Ley, Loreti, and Nguyen for some special cases, independently, but the general
cases remain widely open. We establish a convergence result to asymptotic
solutions as time goes to infinity under rather general assumptions by using
dynamical properties of value functions.
