Stochastic approximation algorithms with constant step size whose average is cooperative

We consider stochastic approximation algorithms with constant step size who se average ordinary differential equation (ODE) is cooperative and irreduci ble. We show that, under mild conditions on the noise process, invariant me asures and empirical occupations measures of the process weakly converge (a s the time goes to infinity and the step size goes to zero) toward measures which are supported by stable equilibria of the ODE. These results are app lied to analyzing the long-term behavior of a class of learning processes a rising in game theory.