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Excited Brownian motions as limits of excited random walks
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  • Published: 27 September 2011

Excited Brownian motions as limits of excited random walks

  • Olivier Raimond1 &
  • Bruno Schapira2 

Probability Theory and Related Fields volume 154, pages 875–909 (2012)Cite this article

  • 208 Accesses

  • 4 Citations

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Abstract

We obtain the convergence in law of a sequence of excited (also called cookies) random walks toward an excited Brownian motion. This last process is a continuous semi-martingale whose drift is a function, say φ, of its local time. It was introduced by Norris, Rogers and Williams as a simplified version of Brownian polymers, and then recently further studied by the authors. To get our results we need to renormalize together the sequence of cookies, the time and the space in a convenient way. The proof follows a general approach already taken by Tóth and his coauthors in multiple occasions, which goes through Ray-Knight type results. Namely we first prove, when φ is bounded and Lipschitz, that the convergence holds at the level of the local time processes. This is done via a careful study of the transition kernel of an auxiliary Markov chain which describes the local time at a given level. Then we prove a tightness result and deduce the convergence at the level of the full processes.

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References

  1. Amir G., Benjamini I., Kozma G.: Excited random walk against a wall. Probab. Theory Relat. Fields 140, 83–102 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arratia, R.A.: Coalescing Brownian motions on the line. Ph.D. Thesis, University of Wisconsin, Madison (1979)

  3. Athreya K., Ney P.: Branching processes. DieGrundlehren der mathematischenWissenschaften,Band 196, pp. xi+287. Springer-Verlag, New York-Heidelberg (1972)

    Google Scholar 

  4. Basdevant A.-L., Singh A.: On the speed of a cookie random walk. Probab. Theory Relat. Fields 141, 625–645 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics, x+277 pp. A Wiley-Interscience Publication. Wiley, New York (1999)

  6. Benjamini, I., Kozma, G., Schapira, Br.: A balanced excited random walk. Preprint. arXiv:1009.0741

  7. Benjamini I., Wilson D.B.: Excited random walk. Electron. Commun. Probab. 8, 86–92 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cranston M., Le Jan Y.: Self-attracting diffusions: two case studies. Math. Ann. 303, 87–93 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Davis B.: Weak limits of perturbed Brownian motion and the equation \({Y_t=B_t+\alpha{\rm sup}\{Y_s : s\le t\} +\beta {\rm inf} \{Y_s : s \le t\}}\) . Ann. Probab. 24, 2007–2023 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dolgopyat, D.: Central limit theorem for excited random walk in the recurrent regime. Preprint. http://www.math.umd.edu/~dmitry/papers.html

  11. Duminil-Copin, H., Smirnov, S.: The connective constant of the honeycomb lattice equals \({\sqrt{2+\sqrt2}}\) . arXiv:1007.0575

  12. Ethier, N., Kurtz, G.: Markov Processes. Characterization and Convergence, x+534 pp. Wiley Series Probab. Math. Stat., New York (1986)

  13. Feller, W.: An Introduction to Probability Theory and its Applications, 2nd edn, vol. II, xxiv+669 pp. Wiley, New York-London-Sydney (1971)

  14. Fontes L.R.G., Isopi M., Newman C.M., Ravishankar K.: The Brownian web: characterization and convergence. Ann. Probab. 32, 2857–2883 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Herrmann S., Roynette B.: Boundedness and convergence of some self-attracting diffusions. Math. Ann. 325, 81–96 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kesten H., Kozlov M.V., Spitzer F.: A limit law for random walk in a random environment. Compositio Math. 30, 145–168 (1975)

    MathSciNet  MATH  Google Scholar 

  17. Kesten, H., Raimond, O., Schapira Br.: Random walks with occasionally modified transition probabilities. arXiv:0911.3886

  18. Kozma, G.: Problem session. In: Oberwolfach report 27/2007, Non-classical interacting random walks. www.mfo.de

  19. Kosygina E., Zerner M.P.W.: Positively and negatively excited random walks on integers, with branching processes. Electron. J. Probab. 13, 1952–1979 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lawler G., Schramm O., Werner W.: On the scaling limit of planar self-avoiding walk. In: (eds) Fractal Geometry and Application. A Jubilee of Benoit Mandelbrot. Proc. Sympos. Pure Math., vol. 72, Part 2, pp. 339–364. Amer. Math. Soc., Providence (2004)

    Google Scholar 

  21. Merkl, F., Rolles, S.W.W.: Linearly edge-reinforced random walks. Dynamics & Stochastics, pp. 66–77. IMS Lecture Notes Monogr. Ser., vol. 48. Inst. Math. Statist., Beachwood (2006)

  22. Menshikov, M., Popov, S., Ramirez, A., Vachkovskaia, M.: On a general many-dimensional excited random walk. arXiv:1001.1741

  23. Norris J.R., Rogers L.C.G., Williams D.: An excluded volume problem for Brownian motion. Phys. Lett. A. 112, 16–18 (1985)

    Article  MathSciNet  Google Scholar 

  24. Norris J.R., Rogers L.C.G., Williams D.: Self-avoiding random walk: a Brownian motion model with local time drift. Probab. Theory Relat. Fields 74, 271–287 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pemantle R.: A survey of random processes with reinforcement. Probab. Surv. 4, 1–79 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pemantle R., Volkov S.: Vertex-reinforced random walk on Z has finite range. Ann. Probab. 27, 1368–1388 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Raimond O.: Self-attracting diffusions: case of the constant interaction. Probab. Theory Relat. Fields 107, 177–196 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Raimond, O., Schapira, Br.: Excited Brownian motion. arXiv:0810.3538

  29. Revuz D., Yor M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)

    Google Scholar 

  30. Sellke T.: Recurrence of reinforced random walk on a ladder. Electron. J. Probab. 11, 301–310 (2006)

    Article  MathSciNet  Google Scholar 

  31. Tarrès P.: Vertex-reinforced random walk on \({\mathbb{Z}}\) eventually gets stuck on five points. Ann. Probab. 32, 2650–2701 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Tóth B.: The “true” self-avoiding walk with bond repulsion on \({\mathbb{Z}}\) : limit theorems. Ann. Probab. 23, 1523–1556 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tóth, B.: Self-interacting random motions. In: Proceedings of the 3rd European Congress of Mathematics, Barcelona 2000, pp. 555–565. Birkhäuser (2001)

  34. Tòth B.: Generalized Ray-Knight theory and limit theorems for self-interacting random walks on Z 1. Ann. Probab. 24, 1324–1367 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tóth B., Werner W.: The true self-repelling motion. Probab. Theory Relat. Fields 111, 375–452 (1998)

    Article  MATH  Google Scholar 

  36. Vervoort, M.R.: Reinforced Random Walks. In preparation. http://staff.science.uva.nl/vervoort/

  37. Werner W.: Some remarks on perturbed reflecting Brownian motion. Sém. Probab. XXIX, LNM 1613, pp. 37–43. Springer, Berlin (1995)

    Google Scholar 

  38. Zerner M.P.W.: Multi-excited random walks on integers. Probab. Theory Relat. Fields 133, 98–122 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire Modal’X, Université Paris Ouest Nanterre La Défense, Bâtiment G, 200 avenue de la République, 92000, Nanterre, France

    Olivier Raimond

  2. Département de Mathématiques, Bât. 425, Université Paris-Sud 11, 91405, Orsay Cedex, France

    Bruno Schapira

Authors
  1. Olivier Raimond
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  2. Bruno Schapira
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Correspondence to Bruno Schapira.

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Raimond, O., Schapira, B. Excited Brownian motions as limits of excited random walks. Probab. Theory Relat. Fields 154, 875–909 (2012). https://doi.org/10.1007/s00440-011-0388-x

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  • Received: 21 October 2010

  • Revised: 11 August 2011

  • Published: 27 September 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0388-x

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Mathematics Subject Classification (2000)

  • 60F17
  • 60K35
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