If we can set up a lyaponuv function for the model, then this system means it can go to equilibrium.
F(x) is a lyaponuv function.
Assumption1: F(x) has a maximum value.
Assumption2:if x(t+1) != x(t) F(x(t+1)) > F(x(t)) + K where K > 0
claim: At some point x(t+1) = x(t)
The time it takes to converge in equalibrium = (maximum state - minimum state) / periodic transistion number;
lyaponuv may come to local optima which means the system will not stay in the maximum or minimum state.
Exchange market is an good example of lyaponuv function process.(without negative externities which means nothing will prevent the increment or decrement)
lyaponuv process end up in stochastic equilibrium comparing to the unique equilibrium in marckov process