Multiple time scale analysis of interacting diffusions - Dawson, Donald Andrew; Greven, Andreas (Probability Theory and Related Fields, 1993)
We consider the questions: how can the long term behavior of large systems of interacting components be described in terms of infinite systems? On what time scale does the infinite system give a qualitatively correct description and what happens at large (resp. critical) time scales? Let Y(t) be a solution (y(t))of the system of stochastic differential equations (w(t) are i.i.d. brownian motions) Mathematical expression In the McKean-Vlasov limit, N→∞, we obtain the infinite independent system Mathematical expression This infinite system has a one parameter set of invariant measures Mathematical expression with γ the equilibrium measure of Mathematical expression. Let Q(·,·) be the transition kernel of the diffusion with generator Mathematical expression with Mathematical expression. Then one main result is that as N→∞ Mathematical expression This provides a new example of a phenomenon also exhibited by the voter model and branching random walk. In particular we are also able to modify our model by adding the term cN(A-y(t))dt to obtain the first example in which the analog of Q(·,·) converges to an honest equilibrium instead of absorption in traps as in all models previously studied in the literature. Finally, we discuss a hierarchical model with two levels from the point of view discussed above but now in two fast time scales. © 1993 Springer-Verla
Keywords: Mathematics Subject Classification: 60K35