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Multiple time scale analysis of interacting diffusions
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  • Published: December 1993

Multiple time scale analysis of interacting diffusions

  • Donald Dawson1 &
  • Andreas Greven2 

Probability Theory and Related Fields volume 95, pages 467–508 (1993)Cite this article

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  • 31 Citations

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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)

$$dy_i^N (t) = \left( {\frac{1}{{2N + 1}}\sum\limits_{j = - N}^N {y_j^N (t) - y_i^N (t)} } \right)dt + \sqrt {2g(y_i^N (t))} dw_i (t).$$

In the McKean-Vlasov limit,N→∞, we obtain the infinite independent system

$$dy_i^\infty (t) = (E(y_i^\infty (t)) - y_i^\infty (t))dt + \sqrt {2g(y_i^\infty (t))} dw_i (t),i \in Z.$$

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→∞

$$\mathcal{L}((Y^N (s(2N + 1)))) \Rightarrow \int {Q_s (\Theta ',d\Theta )v_\Theta ,\Theta '} = E(y_0 ).$$

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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References

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Authors and Affiliations

  1. Department of Mathematics & Statistics, Carleton University, K1S 5B6, Ottawa, Canada

    Donald Dawson

  2. Institut für Mathematische Stochastik, Universität Göttingen, Lotzestrasse 13, W-3400, Göttingen, Germany

    Andreas Greven

Authors
  1. Donald Dawson
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  2. Andreas Greven
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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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  • Received: 24 April 1990

  • Revised: 01 September 1992

  • Issue Date: December 1993

  • DOI: https://doi.org/10.1007/BF01196730

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