Dynamics of Morse-Smale urn processes

We consider stochastic processes {x(n)}(n greater than or equal to 0) of the form x(n+1)-x(n)=gamma(n+1)(F(x(n))+U-n+1) where F : R(m) --> R(m) is C-2, {gamma(i)}(i greater than or equal to 1) is a Sequence of positive numbers decreasing to 0 and {U-i}(i greater than or equal to 1) is a sequence of uniformly bounded R(m)-valued random variables forming suitable martingale differences. We show that when the vector field F is Morse-Smale, almost surely every sample path approaches an asymptotically stable periodic orbit of the deterministic dynamical system dy/dt = F(y). In the case of certain generalized urn processes we show that for each such orbit Gamma, the probability of sample paths approaching Gamma is positive. This gives the generic behavior of three-color urn models.