Communications in Mathematical Physics

, Volume 135, Issue 2, pp 355–378 | Cite as

Scaling limit for interacting Ornstein-Uhlenbeck processes

  • Stefano Olla
  • S. R. S. Varadhan


The problem of describing the bulk behavior of an interacting system consisting of a large number of particles comes up in different contexts. See for example [1] for a recent exposition. In [4] one of the authors considered the case of interacting diffusions on a circle and proved that the density of particles evolves according to a nonlinear diffusion equation. The interacting particles evolved according to a generator that was symmetric in equilibrium. In this article we consider interacting Ornstein-Uhlenbeck processes. Here the diffusion generator is not symmetric relative to the equilibrium and the earlier methods have to be modified considerably. We use some ideas that were employed in [3] to extend the central limit theorem from the symmetric to nonsymmetric cases.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Limit Theorem 
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  1. 1.
    Deuschel, J.D., Stroock, D.W.: Large deviations. New York: Academic Press 1989Google Scholar
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    De Masi, A., Presutti, E.: Lectures on the collective behavior of particle systems. C.A.R.R. Reports in Mathematical Physics, 5/89, 1989Google Scholar
  3. 3.
    Papanicolaou, G., Varadhan, S.R.S.: Ornstein-Uhlenbeck process in a random potential. Commun. Pure Appl. Math.35, 819–834 (1985)Google Scholar
  4. 4.
    Varadhan, S.R.S.: Scaling limits for interacting diffusions. Commun. Math. Phys.135, 313–353 (1991)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Stefano Olla
    • 1
  • S. R. S. Varadhan
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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