Let V be a countably infinite set, and let {Xn,n=0,1,...} be random vectors in ℝV which satisfy Xn=AnXnÅ-1+Sωn, for i.i.d. random matrices {An} and i.i.d. random vectors {Sωn}. Interpretation: site x in V is occupied by Xn(x) particles at time n; An describes random transport of existing particles, and Sωn(x) is the number of ‘births’ at x. Conditions are given for (1) convergence of the sequence {Xn} to equilibrium, and (2) a central limit theorem for n’-1’/2(X1+ëëë+Xn), respectively. When the matrices {An} consist of 0’s and 1’s, these conditions are checked in two classes of examples: the ‘drip, stick and flow model’ (a stochastic flow with births), and a neural network model.
