Advertisement

Probability Theory and Related Fields

, Volume 171, Issue 3–4, pp 865–915 | Cite as

Explicit LDP for a slowed RW driven by a symmetric exclusion process

Article
  • 49 Downloads

Abstract

We consider a random walk (RW) driven by a simple symmetric exclusion process (SSE). Rescaling the RW and the SSE in such a way that a joint hydrodynamic limit theorem holds we prove a joint path large deviation principle. The corresponding large deviation rate function can be split into two components, the rate function of the SSE and the one of the RW given the path of the SSE. These components have different structures (Gaussian and Poissonian, respectively) and to overcome this difficulty we make use of the theory of Orlicz spaces. In particular, the component of the rate function corresponding to the RW is explicit.

Keywords

Large deviations Random environments Hydrodynamic limits Particle systems Exclusion process 

Mathematics Subject Classification

60F10 82C22 82D30 

Notes

Acknowledgements

L.A. has been supported by NWO Gravitation Grant 024.002.003-NETWORKS. M.J. has been partially supported by the ERC Horizon 2020 grant #715734 and by NWO Gravitation Grant 024.002.003-NETWORKS.

References

  1. 1.
    Avena, L., Franco, T., Jara, M., Völlering, F.: Symmetric exclusion as a random environment: hydrodynamic limits. Ann. Inst. H. Poincaré Probab. Statist. 51(3), 901–916 (2015)Google Scholar
  2. 2.
    Avena, L., den Hollander, F., Redig, F.: Large deviation principle for one-dimensional random walk in dynamic random environment: attractive spin-flips and simple symmetric exclusion. Markov Proc. Relat. Fields 16, 139–168 (2010)MathSciNetMATHGoogle Scholar
  3. 3.
    Avena, L., dos Santos, R., Völlering, F.: A transient random walk driven by an exclusion process: regenerations, limit theorems and an Einstein relation. Lat. Am. J. Prob. Math. Stat. (ALEA) 10(2), 693–709 (2013)MATHGoogle Scholar
  4. 4.
    Avena, L., Thomann, P.: Continuity and anomalous fluctuations in random walks in dynamic random environments: numerics, phase diagrams and conjectures. J. Stat. Phys. 147, 1041–1067 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bodineau, T., Lagouge, M.: Large deviations of the empirical currents for a boundary driven reaction diffusion model. Ann. Appl. Probab. 22(6), 2282–2319 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Campos, D., Drewitz, A., Ramirez, A.F., Rassoul-Agha, F., Seppalainen, T.: Level 1 quenched large deviation principle for random walk in dynamic random environment. Spec. Issue Bull. Inst. Math. Acad. Sin. (N.S.) in honor of the 70th birthday of S.R.S. Varadhan 8(1), 1–29 (2013)Google Scholar
  7. 7.
    Comets, F., Gantert, N., Zeitouni, O.: Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab. Theory Relat. Fields 118, 65–114 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications Applications of Mathematics, vol. 38. Springer, New York (1998)CrossRefMATHGoogle Scholar
  9. 9.
    Donsker, M.D., Varadhan, S.R.S.: Large deviations from a hydrodynamic scaling limit. Commun. Pure Appl. Math. 42(3), 243–270 (1989)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gonçalves, P., Jara, M.: Scaling limits of additive functionals of interacting particle systems. Commun. Pure Appl. Math. 66(5), 649–677 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Greven, A., den Hollander, F.: Large deviations for a random walk in random environment. Ann. Probab. 22, 1381–1428 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Guo, M.Z., Papanicolau, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118, 31–59 (1988)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Harris, T.E.: Diffusion with “collision” between particles. J. Appl. Probab. 2, 323–338 (1965)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hilario, M., den Hollander, F., Sidoravicius, V., dos Santos, R., Texeira, A.: Random walk on random walks. Electron. J. Probab. 20(95), 1–35 (2015)MathSciNetMATHGoogle Scholar
  15. 15.
    den Hollander, F., dos Santos, R.: Scaling of a random walk on a supercritical contact process. Ann. Inst. Henri Poincaré Stat. 50(4), 1276–1300 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Huveneers, F., Simenhaus, F.: Random walk driven by simple exclusion process. Electron. J. Probab. 20(105), 1–42 (2015)MathSciNetMATHGoogle Scholar
  17. 17.
    Ignatiouk-Robert, I.: Large deviations for a random walk in dynamical random environment. Ann. Inst. Henri Poincaré 34, 601–636 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jara, M., Landim, C., Sethuraman, S.: Nonequilibrium fluctuations for a tagged particle in mean-zero one dimensional zero-range processes. Probab. Theory Relat. Fields 145, 565–590 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kipnis, C., Landim, C.: Scaling Limits of Particle Systems Grundlehren der Mathematischen Wissenschaften, vol. 320. Springer, Berlin (1999)MATHGoogle Scholar
  20. 20.
    Kipnis, C., Olla, S., Varadhan, S.R.S.: Hydrodynamics and large deviation for simple exclusion processes. Commun. Pure Appl. Math. 42(2), 115–137 (1989)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion. Commun. Math. Phys. 104(1), 1–19 (1986)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Krasnoselskii, M.A., Rutickii, J.B.: Convex Functions and Orlicz Spaces, translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen (1961)Google Scholar
  23. 23.
    Peres, Y., Stauffer, A., Steif, J.E.: Random walks on dynamical percolation: mixing times, mean squared displacement and hitting times. Probab. Theory Relat. Fields 162, 487–530 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rassoul-Agha, F., Seppalainen, T., Yilmaz, A.: Quenched free energy and large deviations for random walks in random potentials. Commun. Pure Appl. Math. 66(2), 202–244 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Redig, F., Völlering, F.: Random walks in dynamic random environments: a transference principle. Ann. Probab. 41(5), 3157–3180 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Sethuraman, S., Varadhan, S.R.S.: Large deviations for the current and tagged particle in 1D nearest-neighbor symmetric simple exclusion. Ann. Probab. 41(3A), 1461–1512 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Skorohod, A.V.: Limit theorems for stochastic processes. Theory Probab. Appl. 1, 261–290 (1956)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Solomon, F.: Random walks in a random environment. Ann. Probab. 3(1), 1–31 (1975)CrossRefMATHGoogle Scholar
  29. 29.
    Spitzer, F.: Interaction of Markov processes. Adv. Math. 5, 246–290 (1970)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Yilmaz, A.: Large deviations for random walk in a space-time product environment. Ann. Probab. 37, 189–205 (2009)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Zeitouni, O.: Random walks in random environments. J. Phys. A Math. Gen. 39, 433–464 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.MIUniversity of LeidenLeidenThe Netherlands
  2. 2.IMPARio de JaneiroBrazil
  3. 3.University of BathBathUK

Personalised recommendations