Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Excited Brownian motions as limits of excited random walks

  • 169 Accesses

  • 3 Citations


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.

This is a preview of subscription content, log in to check access.


  1. 1

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

  2. 2

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

  3. 3

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

  4. 4

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

  5. 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. 6

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

  7. 7

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

  8. 8

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

  9. 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)

  10. 10

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

  11. 11

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

  12. 12

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

  13. 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. 14

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

  15. 15

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

  16. 16

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

  17. 17

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

  18. 18

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

  19. 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)

  20. 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)

  21. 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. 22

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

  23. 23

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

  24. 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)

  25. 25

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

  26. 26

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

  27. 27

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

  28. 28

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

  29. 29

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

  30. 30

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

  31. 31

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

  32. 32

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

  33. 33

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

  34. 34

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

  35. 35

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

  36. 36

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

  37. 37

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

  38. 38

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

Download references

Author information

Correspondence to Bruno Schapira.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Mathematics Subject Classification (2000)

  • 60F17
  • 60K35