Continuous-time random walks with reset events

Historical background and new perspectives
  • Miquel Montero
  • Axel Masó-Puigdellosas
  • Javier Villarroel
Regular Article
Part of the following topical collections:
  1. Topical issue: Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook

Abstract

In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples.

References

  1. 1.
    E.W. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965)ADSCrossRefGoogle Scholar
  2. 2.
    M.F. Shlesinger, J. Stat. Phys. 10, 421 (1974)ADSCrossRefGoogle Scholar
  3. 3.
    B.D. Hughes, E.W. Montroll, M.F. Shlesinger, J. Stat. Phys. 28, 111 (1982)ADSCrossRefGoogle Scholar
  4. 4.
    G.H. Weiss, J.M. Porrà, J. Masoliver, Phys. Rev. E 58, 6431 (1998)ADSCrossRefGoogle Scholar
  5. 5.
    G. Margolin, B. Berkowitz, Phys. Rev. E 65, 031101 (2002)ADSCrossRefGoogle Scholar
  6. 6.
    A. Helmstetter, D. Sornette, Phys. Rev. E 66, 061104 (2002)ADSCrossRefGoogle Scholar
  7. 7.
    M.S. Mega, P. Allegrini, P. Grigolini, V. Latora, L. Palatella, Phys. Rev. Lett. 90, 188501 (2003)ADSCrossRefGoogle Scholar
  8. 8.
    B. Berkowitz, H. Scher, Phys. Rev. Lett. 79, 4038 (1997)ADSCrossRefGoogle Scholar
  9. 9.
    M. Boguñá, Á. Corral, Phys. Rev. Lett. 78, 4950 (1997)ADSCrossRefGoogle Scholar
  10. 10.
    J. Masoliver, M. Montero, G.H. Weiss, Phys. Rev. E 67, 021112 (2003)ADSCrossRefGoogle Scholar
  11. 11.
    J. Masoliver, M. Montero, J. Perelló, Phys. Rev. E 71, 056130 (2005)ADSCrossRefGoogle Scholar
  12. 12.
    M. Montero, J. Perelló, J. Masoliver, F. Lillo, S. Micciché, R.N. Mantegna, Phys. Rev. E 72, 056101 (2005)ADSCrossRefGoogle Scholar
  13. 13.
    J. Masoliver, M. Montero, J. Perelló, G.H. Weiss, J. Econ. Behav. Organ. 61, 577 (2006)CrossRefGoogle Scholar
  14. 14.
    E. Scalas, Physica A 362, 225 (2006)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    V. Méndez, D. Campos, F. Bartumeus, in Stochastic Foundations in Movement Ecology (Springer-Verlag, Berlin, 2014).Google Scholar
  16. 16.
    F. Höfling, T. Franosch, Rep. Prog. Phys. 76, 046602 (2013)ADSCrossRefGoogle Scholar
  17. 17.
    B. Levikson, J. Appl. Probab. 14, 492 (1977)MathSciNetCrossRefGoogle Scholar
  18. 18.
    A.G. Pakes, J. Appl. Probab. 15, 65 (1978)ADSCrossRefGoogle Scholar
  19. 19.
    A.G. Pakes, Stoc. Proc. Appl. 8, 277 (1979)CrossRefGoogle Scholar
  20. 20.
    E.G. Kyriakidis, Stat. Probab. Lett. 20, 239 (1994)CrossRefGoogle Scholar
  21. 21.
    R.J. Swift, Int. J. Math. Math. Sci. 25, 689 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    X. Chao, Y. Zheng, Probab. Eng. Inform. Sci. 17, 83 (2003)CrossRefGoogle Scholar
  23. 23.
    B. Krishna Kumar, D. Arivudainambi, Comput. Math. Appl. 40, 1233 (2000)MathSciNetCrossRefGoogle Scholar
  24. 24.
    A. Di Crescenzo, V. Giorno, A.G. Nobile, L.M. Ricciardi, Queuing Syst. 43, 329 (2003)CrossRefGoogle Scholar
  25. 25.
    M. Luby, A. Sinclair, D. Zuckerman, Inf. Process. Lett. 47, 173 (1993)CrossRefGoogle Scholar
  26. 26.
    S.C. Manrubia, D.H. Zanette, Phys. Rev. E 59, 4945 (1999)ADSCrossRefGoogle Scholar
  27. 27.
    M.R. Evans, S.N. Majumdar, Phys. Rev. Lett. 106, 160601 (2011)ADSCrossRefGoogle Scholar
  28. 28.
    M.R. Evans, S.N. Majumdar, J. Phys. A: Math. Theor. 44, 435001 (2011)ADSCrossRefGoogle Scholar
  29. 29.
    J. Whitehouse, M.R. Evans, S.N. Majumdar, Phys. Rev. E 87, 022118 (2013)ADSCrossRefGoogle Scholar
  30. 30.
    M.R. Evans, S.N. Majumdar, J. Phys. A: Math. Theor. 47, 285001 (2014)CrossRefGoogle Scholar
  31. 31.
    A. Pal, Phys. Rev. E 91, 012113 (2015)ADSCrossRefGoogle Scholar
  32. 32.
    A. Pal, A. Kundu, M.R. Evans, J. Phys. A: Math. Theor. 49, 225001 (2016)ADSCrossRefGoogle Scholar
  33. 33.
    S. Eule, J.J. Metzger, New J. Phys. 18, 033006 (2016)ADSCrossRefGoogle Scholar
  34. 34.
    J.M. Meylahn, S. Sabhapandit, H. Touchette, Phys. Rev. E 92, 062148 (2015)ADSCrossRefGoogle Scholar
  35. 35.
    U. Bhat, D. De Bacco, S. Redner, J. Stat. Mech.: Theory Exp. 2016, 083401 (2016)CrossRefGoogle Scholar
  36. 36.
    D. Boyer, M.R. Evans, S.N. Majumdar, J. Stat. Mech.: Theory Exp. 2017, 023208 (2017)CrossRefGoogle Scholar
  37. 37.
    S. Reuveni, Phys. Rev. Lett. 116, 170601 (2016)ADSCrossRefGoogle Scholar
  38. 38.
    A. Pal, S. Reuveni, Phys. Rev. Lett. 118, 030603 (2017)ADSCrossRefGoogle Scholar
  39. 39.
    R. Falcao, M.R. Evans, J. Stat. Mech.: Theory Exp. 2017, 023204 (2017)CrossRefGoogle Scholar
  40. 40.
    X. Durang, M. Henkel, H. Park, J. Phys. A: Math. Theor. 47, 045002 (2014)ADSCrossRefGoogle Scholar
  41. 41.
    T. Rotbart, S. Reuveni, M. Urbakh, Phys. Rev. E 92, 060101(R) (2015)ADSCrossRefGoogle Scholar
  42. 42.
    É. Roldán, A. Lisica, D. Sánchez-Taltavull, S.W. Grill, Phys. Rev. E 93, 062411 (2016)ADSCrossRefGoogle Scholar
  43. 43.
    J. Fuchs, S. Goldt, U. Seifert, Europhys. Lett. 113, 60009 (2016)ADSCrossRefGoogle Scholar
  44. 44.
    D. Boyer, C. Solis-Salas, Phys. Rev. Lett. 112, 240601 (2014)ADSCrossRefGoogle Scholar
  45. 45.
    Ł. Kuśmierz, S.N. Majumdar, S. Sabhapandit, G. Schehr, Phys. Rev. Lett. 113, 220602 (2014)ADSCrossRefGoogle Scholar
  46. 46.
    S.N. Majumdar, S. Sabhapandit, G. Schehr, Phys. Rev. E 92, 052126 (2015)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    M. Montero, J. Villarroel, Phys. Rev. E 94, 032132 (2016)ADSCrossRefGoogle Scholar
  48. 48.
    M. Montero, J. Villarroel, Phys. Rev. E 87, 012116 (2013)ADSCrossRefGoogle Scholar
  49. 49.
    Ł. Kuśmierz, E. Gudowska-Nowak, Phys. Rev. E 92, 052127 (2015)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    D. Campos, V. Méndez, Phys. Rev. E 92, 062115 (2015)ADSCrossRefGoogle Scholar
  51. 51.
    V. Méndez, D. Campos, Phys. Rev. E 93, 022106 (2016)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Departament de Física de la Matèria Condensada, Universitat de Barcelona (UB)BarcelonaSpain
  2. 2.Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de BarcelonaBarcelonaSpain
  3. 3.Departamento de Matemáticas & Instituto Universitario de Física Fundamental y Matemáticas, Universidad de SalamancaSalamancaSpain

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