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Random Walk Among Mobile/Immobile Traps: A Short Review

  • Siva Athreya
  • Alexander Drewitz
  • Rongfeng SunEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 300)

Abstract

There have been extensive studies of a random walk among a field of immobile traps (or obstacles), where one is interested in the probability of survival as well as the law of the random walk conditioned on its survival up to time t. In contrast, very little is known when the traps are mobile. We will briefly review the literature on the trapping problem with immobile traps, and then review some recent results on a model with mobile traps, where the traps are represented by a Poisson system of independent random walks on \(\mathbb {Z}^d\). Some open questions will be given at the end.

Keywords

Trapping problem Parabolic anderson model Random walk in random potential 

Notes

Acknowledgement

R.S. is supported by NUS grant R-146-000-220-112. S.A. is supported by CPDA grant and ISF-UGC project. We thank Ryoki Fukishima for helpful comments that corrected some earlier misstatements. Lastly, we thank Vladas Sidoravicius for encouraging us to write this review, and we are deeply saddened by his untimely death.

References

  1. 1.
    Antal, P.: Enlargement of obstacles for the simple random walk. Ann. Probab. 23, 1061–1101 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Athreya, S., Drewitz, A., Sun, R.: Subdiffusivity of a random walk among a Poisson system of moving traps on \(\mathbb{Z}\). Math. Phys. Anal. Geom. 20, 1 (2017). Art. 1MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bolthausen, E.: Localization of a two-dimensional random walk with an attractive path interaction. Ann. Probab. 22, 875–918 (1994)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borrego, R., Abad, E., Yuste, S.B.: Survival probability of a subdiffusive particle in a d-dimensional sea of mobile traps Phys. Rev. E 80, 061121 (2009)Google Scholar
  5. 5.
    Biskup, M., König, W.: Eigenvalue order statistics for random Schrödinger operators with doubly exponential tails. Commun. Math. Phys. 341, 179–218 (2016)CrossRefGoogle Scholar
  6. 6.
    Biskup, M., König, W., dos Santos, R.: Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails. Probab. Theory Relat. Fields. arXiv 1609.00989 (2016). To appear inGoogle Scholar
  7. 7.
    Chen, L.C., Sun, R.: A monotonicity result for the range of a perturbed random walk. J. Theoret. Probab. 27, 997–1010 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Drewitz, A., Gärtner, J., Ramírez, A.F., Sun, R.: Survival probability of a random walk among a Poisson system of moving traps. In: DProbability in Complex Physical Systems—In honour of Erwin Bolthausen and Jürgen Gärtner. Springer Proceedings in Mathematics, vol. 11, pp. 119–158 (2012)zbMATHGoogle Scholar
  9. 9.
    Drewitz, A., Sousi, P., Sun, R.: Symmetric rearrangements around infinity with applications to Lévy processes. Probab. Theory Relat. Fields 158(3–4), 637–664 (2014)CrossRefGoogle Scholar
  10. 10.
    Donsker, M.D., Varadhan, S.R.S.: Asymptotics for the Wiener sausage. Commun. Pure Appl. Math. 28(4), 525–565 (1975)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Donsker, M.D., Varadhan, S.R.S.: On the number of distinct sites visited by a random walk. Commun. Pure Appl. Math. 32(6), 721–747 (1979)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ding, J., Xu, C.: Poly-logarithmic localization for random walks among random obstacles. Ann. Probab. arXiv:1703.06922 (2017). To appear in
  13. 13.
    Fukushima, R.: From the Lifshitz tail to the quenched survival asymptotics in the trapping problem. Electron. Commun. Probab. 14, 435–446 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gärtner, J., den Hollander, F., Maillard, G.: Intermittency on catalysts. In: Trends in Stochastic Analysis. London Mathematical Society Lecture Note Series, vol. 353, pp. 235–248. Cambridge University Press, Cambridge (2009)Google Scholar
  15. 15.
    Gärtner, J., den Hollander, F., Maillard, G.: Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment. In: Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol. 11, pp. 159–193. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  16. 16.
    van der Hofstad, R., König, W., Mörters, P.: The universality classes in the parabolic Anderson model. Commun. Math. Phys. 267(2), 307–353 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    den Hollander, F., Weiss, G.H.: Aspects of trapping in transport processes. In: Contemporary Problems in Statistical Physics, SIAM, Philadelphia (1994)Google Scholar
  18. 18.
    Komorowski, T.: Brownian motion in a Poisson obstacle field. Astérisque 266, 91–111 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    König, W.: The parabolic Anderson model. In: Random Walk in Random Potential. Pathways in Mathematics. Birkhäuser/Springer (2016)Google Scholar
  20. 20.
    Mörters, P.: The parabolic Anderson model with heavy-tailed potential. In: Surveys in Stochastic Processes, pp. 67–85. EMS Series of Congress Reports. European Mathematical Society, Zürich (2011)Google Scholar
  21. 21.
    Moreau, M., Oshanin, G., Bénichou, O., Coppey, M.: Pascal principle for diffusion-controlled trapping reactions. Phys. Rev. E 67, 045104(R) (2003)CrossRefGoogle Scholar
  22. 22.
    Moreau, M., Oshanin, G., Bénichou, O., Coppey, M.: Lattice theory of trapping reactions with mobile species. Phys. Rev. E 69, 046101 (2004)CrossRefGoogle Scholar
  23. 23.
    Öz, M.: Subdiffusivity of Brownian motion among a Poissonian field of moving traps. ALEA, Lat. Am. J. Probab. Math. Stat. 16, 33–47 (2019)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Povel, T.: Confinement of Brownian motion among Poissonian obstacles in \({ R}^d, d\ge 3\) Probab. Theory Relat. Fields 114(2), 177–205 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Peres, Y., Sinclair, A., Sousi, P., Stauffer, A.: Mobile geometric graphs: detection, coverage and percolation. Probab. Theory Relat. Fields 156, 273–305 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Redig, F.: An exponential upper bound for the survival probability in a dynamic random trap model. J. Stat. Phys. 74, 815–827 (1994)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Schmock, U.: Convergence of the normalized one-dimensional Wiener sausage path measures to a mixture of Brownian taboo processes. Stochast. Stochast. Rep. 29(2), 171–183 (1990)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sznitman, A.-S.: On the confinement property of two-dimensional Brownian motion among Poissonian obstacles. Commun. Pure Appl. Math. 44(8–9), 1137–1170 (1991)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sznitman, A.-S.: Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics. Springer-Verlag, Berlin (1998)CrossRefGoogle Scholar
  30. 30.
    Sethuraman, S.: Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment. Stochast. Process. Appl. 103(2), 169–209 (2003)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Schnitzler, A., Wolff, T.: Precise asymptotics for the parabolic Anderson model with a moving catalyst or trap. In: Probability in Complex Physical Systems—In honour of Erwin Bolthausen and Jürgen Gärtner. Springer Proceedings in Mathematics, vol. 11, pp. 69–89 (2012)Google Scholar
  32. 32.
    Yuste, S.B., Oshanin, G., Lindenberg, K., Bénichou, O., Klafter, J.: Survival probability of a particle in a sea of mobile traps: a tale of tails. Phys. Rev. E 78, 021105 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Indian Statistical InstituteBangaloreIndia
  2. 2.Mathematisches InstitutUniversität zu KölnKölnGermany
  3. 3.Department of MathematicsNational University of SingaporeSingaporeSingapore

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