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Equilibrium Fluctuations of Additive Functionals of Zero-Range Models

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 75))

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Abstract

For mean-zero and asymmetric zero-range processes on \({\mathbb{Z}}^{d}\), the fluctuations of additive functionals starting from an invariant measure are considered. Under certain assumptions, we establish when the fluctuations are diffusive and satisfy functional central limit theorems. These results complement those for symmetric zero-range systems and also those for simple exclusion models already in the literature.

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References

  1. Andjel, E.D.: Invariant measures for the zero range process. Ann. Probab. 10, 525–547 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Evans, M.R., Hanney, T.: Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A: Math. Gen. 38, 195–240 (2005)

    Article  MathSciNet  Google Scholar 

  3. Kipnis, C.: Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab. 14, 397–408, (1986)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften, vol. 320. Springer, Berlin (1999)

    Google Scholar 

  5. Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes. Commun. Math. Phys. 104, 1–19 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  6. Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov Processes. Springer, New York (2012)

    Book  MATH  Google Scholar 

  7. Landim, C., Sethuraman, S., Varadhan, S.: Spectral gap for zero-range dynamics. Ann. Probab. 24, 1871–1902 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Liggett, T.: Interacting Particle Systems. Springer, New York (1985)

    Book  MATH  Google Scholar 

  9. Morris, B.: Spectral gap for the zero-range process with constant rate. Ann. Probab. 34, 1645–1664 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Nagahata, Y.: Spectral gap for zero-range processes with jump rate g(x) = x γ. Stoch. Process. Appl. 120, 949–958 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Quastel, J.: Diffusion of color in the simple exclusion process. Commun. Pure Appl. Math. 45, 623–679 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Quastel, J., Jankowski, H., Sheriff, J.: Central limit theorem for zero-range processes. special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday. Methods Appl. Anal. 9(3), 393–406 (2002)

    Google Scholar 

  13. Sethuraman, S.: Central limit theorems for additive functionals of the simple exclusion process. Ann. Probab. 28, 277–302 (2000); Correction (2006) 34, 427–428

    Google Scholar 

  14. Sethuraman, S.: On extremal measures for conservative particle systems. Ann. Inst. Henri Poincare Prob. et Statistiques 37, 139–154 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Sethuraman, S., Xu, L.: A central limit theorem for reversible exclusion and zero-range particle systems. Ann. Probab. 24, 1842–1870 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Varadhan, S.R.S.: Self-diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion. Ann. Inst. Henri Poincare Prob. et Statistiques. 31, 273–285 (1995)

    MATH  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the enjoyable hospitality of CMAT (Portugal) where the conference “Particle systems and Partial Differential Equations” December 5–7, 2012 took place and part of this work was discussed.

C.B. acknowledges the support of the French Ministry of Education through the grant ANR-10-BLAN 0108 (SHEPI).

P.G. thanks FCT (Portugal) for support through the research project “Non-Equilibrium Statistical Physics” PTDC/MAT/109844/2009. P.G. thanks the Research Centre of Mathematics of the University of Minho, for the financial support provided by “FEDER” through the “Programa Operacional Factores de Competitividade COMPETE” and by FCT through the research project PEst-C/MAT/UI0013/2011.

C.B. and P.G. are grateful to Égide and FCT for the research project FCT/1560/25/1/2012/S.

The research of S.S. was supported in part by NSF DMS-1159026.

Author information

Authors and Affiliations

  1. Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis, UMR CNRS 7351, Parc Valrose, 06108, Nice cedex 02, France

    Cédric Bernardin

  2. Departamento de Matemática, PUC-RIO, Rua Marquês de São Vicente, no. 225, 22453-900, Gávea, Rio de Janeiro, Brazil

    Patrícia Gonçalves

  3. CMAT, Centro de Matemática da Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Patrícia Gonçalves

  4. Mathematics, University of Arizona, 617 N. Santa Rita Ave., Tucson, AZ, 85721, USA

    Sunder Sethuraman

Authors
  1. Cédric Bernardin
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  2. Patrícia Gonçalves
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  3. Sunder Sethuraman
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Corresponding author

Correspondence to Cédric Bernardin .

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Editors and Affiliations

  1. UMR CNRS-UNS 7351, Université de Nice Sophia Antipolis, Laboratoire J.A.Dieudonné, Nice, France

    Cédric Bernardin

  2. Universidade do Minho, Centre of Mathematics, Braga, Portugal

    Patricia Gonçalves

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Bernardin, C., Gonçalves, P., Sethuraman, S. (2014). Equilibrium Fluctuations of Additive Functionals of Zero-Range Models. In: Bernardin, C., Gonçalves, P. (eds) From Particle Systems to Partial Differential Equations. Springer Proceedings in Mathematics & Statistics, vol 75. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54271-8_5

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