Summary
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?
LetY N(t) be a solution (y Ni (t))i∈[−N,N]of the system of stochastic differential equations (w i (t) are i.i.d. brownian motions)
In the McKean-Vlasov limit,N→∞, we obtain the infinite independent system
This infinite system has a one parameter set of invariant measures\(v_\Theta = \mathop \otimes \limits_{x \in Z} \Gamma _\Theta \) with Γθ the equilibrium measure of\(dx(t) = (\Theta - x(t))dt + \sqrt {2g(x(t))} dw(t)\). LetQ s(·,·) be the transition kernel of the diffusion with generator\(u_g (x)\left( {\frac{\partial }{{\partial x}}} \right)^2 \) with\(u_g (x) = \int {g(y)\Gamma x(dy)} \). Then one main result is that asN→∞
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 termcN −1(A−y Ni (t))dt to obtain the first example in which the analog ofQ s (·,·) 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.
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Dawson, D., Greven, A. Multiple time scale analysis of interacting diffusions. Probab. Th. Rel. Fields 95, 467–508 (1993). https://doi.org/10.1007/BF01196730
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DOI: https://doi.org/10.1007/BF01196730