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System of interacting particles and nonlinear diffusion reflecting in a domain with sticky boundary
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  • Published: June 1989

System of interacting particles and nonlinear diffusion reflecting in a domain with sticky boundary

  • Carl Graham1 &
  • Michel Métivier1 

Probability Theory and Related Fields volume 82, pages 225–240 (1989)Cite this article

  • 190 Accesses

  • 7 Citations

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Summary

We study a system of particles and the nonlinear McKean-Vlasov diffusion that is its limit for weak interactions in Statistical Mechanics, reflecting in a domain with sticky boundary. The interaction takes place in particular in the sojourn condition. We show existence and uniqueness for the nonlinear martingale problem, by a contraction argument on time-change. Then we construct the system of particles by a limiting procedure, and show propagation of chaos towards the nonlinear diffusion.

Résumé

Nous étudions un système de particules et la diffusion non-linéaire de type McKean-Vlasov qui en est la limite en Mécanique Statistique pour des interactions faibles, en réflexion dans un domaine à bord collant. L'interaction réside en particulier dans la condition de séjour. Nous montrons l'existence et l'unicité pour le problème de martingales non-linéaire, par une méthode de contraction sur le changement de temps. Nous construisons le système de particules en tant que limite en loi, et démontrons la propagation du chaos vers la diffusion non-linéaire.

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References

  1. Aldous, A.: Exchangeability and related topics. Ecole d'Eté de Saint Flour 1983. Lect. Notes Math., vol. 1117. Berlin Heidelberg New York: Springer 1983

    Google Scholar 

  2. Anderson, R.F., Orey, S.: Small random perturbations of dynamical systems with reflecting boundary. Nagoya Math. J.60, 186–216 (1976)

    Google Scholar 

  3. Bensoussan, A., Lions, J.-L.: Contrôle impulsionnel et inéquations quasi-variationnelles. Paris: Dunod 1982

    Google Scholar 

  4. El Karoui, N.: Processus de diffusion associé à un opérateur élliptique dégénéré et à une condition frontière. Thèse d'Etat, Paris-VI, 1971

  5. Graham, C.: Systèmes de Particules en interaction dans un domaine à paroi collante et problèmes de martingales avec réflexion. Thèse de Docteur 3e cycle, Paris-VI, 1985

  6. Graham, C.: The martingale problem with sticky reflection conditions, and a system of particles interacting at the boundary. Ann. Inst. Henri Poincaré, Prob. Stat.24, 45–72 (1988)

    Google Scholar 

  7. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North Holland 1981

    Google Scholar 

  8. Lions, P.-L., Sznitman, A.-S.: Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math.37, 511–537 (1984)

    Google Scholar 

  9. Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with boundary conditions. Comm. Pure Appl. Math.24, 147–225 (1971)

    Google Scholar 

  10. Sznitman, A.-S.: Nonlinear reflecting diffusion processes, and the propagation of chaos and fluctuations associated. J.F.A.56, 311–336 (1984)

    Google Scholar 

  11. Szulga, A.: On minimal metrics in the space of random variables. Theory Probab. Appl.27, 424–430 (1982)

    Google Scholar 

  12. Weinryb, S.: Théorèmes limites pour certains processus de Markov et études asymptotiques relatives à la saucisse de Wiener. Thèse d'Etat, Paris-VI, 1987

  13. Zolotarev, V.M.: Probability metrics. Theory Probab. Appl.28, 278–302 (1983)

    Google Scholar 

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Authors and Affiliations

  1. Ecole Polytechnique (URA CNRS 756), CMAP, F-91128, Palaiseau, France

    Carl Graham & Michel Métivier

Authors
  1. Carl Graham
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  2. Michel Métivier
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Graham, C., Métivier, M. System of interacting particles and nonlinear diffusion reflecting in a domain with sticky boundary. Probab. Th. Rel. Fields 82, 225–240 (1989). https://doi.org/10.1007/BF00354761

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  • Received: 01 March 1988

  • Revised: 06 March 1989

  • Issue Date: June 1989

  • DOI: https://doi.org/10.1007/BF00354761

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Keywords

  • Stochastic Process
  • Probability Theory
  • Statistical Mechanics
  • Weak Interaction
  • Mathematical Biology
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