Capture Problems For Coupled Random Walks

  • Maury Bramson
  • David Griffeath
Part of the Progress in Probability book series (PRPR, volume 28)


N predator random walks X k (t) stalk a prey random walk Y(t). By conspiring among themselves and observing their prey, the predators try to minimize the capture time T:
$$T = \min \{ t:{X_k}(t) = Y(t)\,{\text{for}}\,{\text{some}}\,k\}.$$


Markov Chain Brownian Motion Random Walk Exit Time Efficient Capture 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    R. Arratia, Limiting point processes for rending of coalescing and annihilating random walks on Z D, Ann. Probab. 9 (1981), 909–936.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    M. Bramson and J.L. Lebowitz, Asymptotic behavior of densities for two-particle annihilating random walks, to appear in J. Stat. Physics.Google Scholar
  3. [3]
    M. Bramson and D. Griffeath, Clustering and dispersion rates for some interacting particle systems on Z1, Ann. Probab. 8 (1980), 183–213.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    D.L. Burkholder, Exit times of Brownian motion, harmonic majorizaLion, and Hardy spaces, Adv. Math. 26 (1977), 182–205.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    R.D. De Blassie, Exit times from cones in Rn of Brownian motion, Z. Wahr. verw. Gebiete 74 (1987), 1–29.Google Scholar
  6. [6]
    E.B. Dynkin and A.A. Yushkevich, Markov Processes: Theorems and Problems, Plenum Press, NY, 1969.Google Scholar
  7. [7]
    D. Griffeath, Coupling methods for Markov processes, Studies in Probability and Ergodic Theory, ed. Rota. Academic Press, New York, NY, 1978, 1–43.Google Scholar
  8. [8]
    T. Lindvall and L.C.G. Rogers, Coupling of multidimensional diffusions by reflection, Ann. Probab. 14 (1986), 860–872.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    T.R. McConnell, Exit times of N-dimensional random walks, Z. Wahr. verw. Gebiete 67 (1984), 213–233.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    F. Spitzer, Some theorems concerning 2-dimensional Brownian motion, Trans. Amer. Math. Soc. 271 (1958), 719–731.MathSciNetGoogle Scholar
  11. [11]
    F. Spitzer, Principles of Random Walk, Graduate Texts in Mathematics 34. Springer-Verlag, New York, 1976.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Maury Bramson
    • 1
  • David Griffeath
    • 1
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA

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