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Independent Particles in a Dynamical Random Environment

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Probability and Analysis in Interacting Physical Systems (VAR75 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 283))

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Abstract

We study the motion of independent particles in a dynamical random environment on the integer lattice. The environment has a product distribution. For the multidimensional case, we characterize the class of spatially ergodic invariant measures. These invariant distributions are mixtures of inhomogeneous Poisson product measures that depend on the past of the environment. We also investigate the correlations in this measure. For dimensions one and two, we prove convergence to equilibrium from spatially ergodic initial distributions. In the one-dimensional situation we study fluctuations of the net current seen by an observer traveling at a deterministic speed. When this current is centered by its quenched mean its limit distributions are the same as for classical independent particles.

F. Rassoul-Agha was partially supported by NSF grant DMS-1407574 and Simons Foundation grant 306576.

M. Joseph and F. Rassoul-Agha were partially supported by NSF grant DMS-0747758.

T. Seppäläinen was partially supported by NSF grants DMS-0701091, DMS-1003651, DMS-1306777 and DMS-1602486, by Simons Foundation grant 338287, and by the Wisconsin Alumni Research Foundation.

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References

  1. Andjel, E.D.: Invariant measures for the zero range processes. Ann. Probab. 10(3), 525–547 (1982)

    Article  MathSciNet  Google Scholar 

  2. Balázs, M., Rassoul-Agha, F., Seppäläinen, T.: The random average process and random walk in a space-time random environment in one dimension. Comm. Math. Phys. 266, 499–545 (2006)

    Article  MathSciNet  Google Scholar 

  3. Barabási, A.L., Stanley, H.E.: Fractal Concepts in Surface Growth. Cambridge University Press (1995)

    Google Scholar 

  4. Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theor. Appl. 1(1), 1130001 (2012)

    Article  MathSciNet  Google Scholar 

  5. Doob, J.L.: Stochastic Processes. Wiley Classics Library. John Wiley & Sons Inc., New York. Reprint of the 1953 original, A Wiley-Interscience Publication (1990)

    Google Scholar 

  6. Durrett, R.: Duxbury advanced series. In: Probability: Theory and Examples, 3rd edn. Brooks/Cole-Thomson, Belmont, CA (2004)

    Google Scholar 

  7. Ekhaus, M., Gray, L.: Convergence to equilibrium and a strong law for the motion of restricted interfaces. Unpublished manuscript (1994)

    Google Scholar 

  8. Ferrari, P.A., Fontes, L.R.G.: Fluctuations of a surface submitted to a random average process. Electron. J. Probab. 3(6), 34 (electronic) (1998)

    Google Scholar 

  9. Jara, M., Peterson, J.: Hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment. Ann. Inst. Henri Poincaré Probab. Stat. 53(4), 1747–1792 (2017)

    Article  MathSciNet  Google Scholar 

  10. Kumar, R.: Space-time current process for independent random walks in one dimension. ALEA Lat. Am. J. Probab. Math. Stat. 4, 307–336 (2008)

    Google Scholar 

  11. Liggett, Thomas M.: Interacting particle systems. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276. Springer, New York (1985)

    Google Scholar 

  12. James, R.: Munkres, 2nd edn. Topology. Prentice-Hall Inc, Upper Saddle River, N.J. (2000)

    Google Scholar 

  13. Norris, J.R.: Markov chains. In: Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2. Cambridge University Press, Cambridge. Reprint of 1997 original (1998)

    Google Scholar 

  14. Peterson, J.: Systems of one-dimensional random walks in a common random environment. Electron. J. Probab. 15(32), 1024–1040 (2010)

    Article  MathSciNet  Google Scholar 

  15. Peterson, J., Seppäläinen, T.: Current fluctuations of a system of one-dimensional random walks in random environment. Ann. Probab. 38(6), 2258–2294 (2010)

    Article  MathSciNet  Google Scholar 

  16. Rassoul-Agha, F., Seppäläinen, T.: An almost sure invariance principle for random walks in a space-time random environment. Probab. Theory Related Fields 133(3), 299–314 (2005)

    Article  MathSciNet  Google Scholar 

  17. Rosenblatt, M.: Markov Processes, Structure and Asymptotic Behavior. Springer, New York. Die Grundlehren der mathematischen Wissenschaften, Band 184 (1971)

    Chapter  Google Scholar 

  18. Seppäläinen, T.: Translation invariant exclusion processes. Lecture notes available at: http://www.math.wisc.edu/~seppalai/excl-book/etusivu.html

  19. Seppäläinen, T.: Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks. Ann. Probab. 33(2), 759–797 (2005)

    Article  MathSciNet  Google Scholar 

  20. Seppäläinen, T.: Current fluctuations for stochastic particle systems with drift in one spatial dimension. Ensaios Matemáticos [Mathematical Surveys], vol. 18. Sociedade Brasileira de Matemática, Rio de Janeiro (2010)

    Google Scholar 

  21. Sethuraman, S.: On extremal measures for conservative particle systems. Ann. Inst. H. Poincaré Probab. Statist. 37(2), 139–154 (2001)

    Article  MathSciNet  Google Scholar 

  22. Spitzer, F.: Graduate texts in mathematics. In: Principles of Random Walks, vol. 34, 2nd edn. Springer, New York (1976)

    Google Scholar 

  23. Stromberg, K.R.: Introduction to Classical Real Analysis. Wadsworth International, Belmont, Calif. Wadsworth International Mathematics Series (1981)

    Google Scholar 

  24. Varadhan, S.R.S.: Probability Theory. Courant Lecture Notes in Mathematics, vol. 7. New York University Courant Institute of Mathematical Sciences, New York (2001)

    Google Scholar 

  25. Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984, vol. 1180 of Lecture Notes in Math., pp. 265–439. Springer, Berlin (1986)

    Google Scholar 

  26. Wheeden, R.L., Zygmund, A.: Pure and applied mathematics. In: Measure and Integral: An Introduction to Real Analysis, vol. 43. Marcel Dekker Inc., New York (1977)

    Book  Google Scholar 

  27. Zygmund, A.: Trigonometric Series. vol. I, II. Cambridge Mathematical Library. Cambridge University Press, Cambridge. Reprint of the 1979 edition (1988)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Indian Statistical Institute, Bangalore, 8th Mile Mysore Road, RVCE Post, Bengaluru, 560059, Karnataka, India

    Mathew Joseph

  2. Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT, 84112, USA

    Firas Rassoul-Agha

  3. Mathematics Department, University of Wisconsin-Madison, Van Vleck Hall 480 Lincoln Dr., Madison, WI, 53706-1388, USA

    Timo Seppäläinen

Authors
  1. Mathew Joseph
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  2. Firas Rassoul-Agha
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  3. Timo Seppäläinen
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Corresponding author

Correspondence to Firas Rassoul-Agha .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, Technische Universität Berlin, Berlin, Germany

    Peter Friz

  2. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Wolfgang König

  3. Institute for Mathematical Stochastics, University of Münster, Münster, Germany

    Chiranjib Mukherjee

  4. CEREMADE, UMR CNRS 7534, Université Paris Dauphine - PSL, Paris, France

    Stefano Olla

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Joseph, M., Rassoul-Agha, F., Seppäläinen, T. (2019). Independent Particles in a Dynamical Random Environment. In: Friz, P., König, W., Mukherjee, C., Olla, S. (eds) Probability and Analysis in Interacting Physical Systems. VAR75 2016. Springer Proceedings in Mathematics & Statistics, vol 283. Springer, Cham. https://doi.org/10.1007/978-3-030-15338-0_4

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