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Survival Probability of a Random Walk Among a Poisson System of Moving Traps

  • Alexander Drewitz
  • Jürgen Gärtner
  • Alejandro F. Ramírez
  • Rongfeng Sun
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 11)

Abstract

We review some old and prove some new results on the survival probability of a random walk among a Poisson system of moving traps on \({\mathbb{Z}}^{d}\), which can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential. We show that the annealed survival probability decays asymptotically as e\({}^{-{\lambda }_{1}\sqrt{t}}\) for d = 1, as e\({}^{-{\lambda }_{2}t/\log t}\) for d = 2, and as e\({}^{-{\lambda }_{d}t}\) for d ≥ 3, where λ1 and λ2 can be identified explicitly. In addition, we show that the quenched survival probability decays asymptotically as e\({}^{-\tilde{{\lambda }}_{d}t}\), with \(\tilde{{\lambda }}_{d} > 0\) for all d ≥ 1. A key ingredient in bounding the annealed survival probability is what is known in the physics literature as the Pascal principle, which asserts that the annealed survival probability is maximized if the random walk stays at a fixed position. A corollary of independent interest is that the expected cardinality of the range of a continuous time symmetric random walk increases under perturbation by a deterministic path.

Keywords

Random Walk Survival Probability Large Deviation Principle Jump Rate Poisson Random Variable 
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.

Notes

Acknowledgements

We thank Frank den Hollander for bringing [22] to our attention, Alain-Sol Sznitman for suggesting that we prove a shape theorem for the quenched survival probability, and Vladas Sidoravicius for explaining to us [16, Prop. 8], which we use to prove the positivity of the quenched Lyapunov exponent. A.F. Ramírez was partially supported by Fondo Nacional de Desarrollo Científico y Tecnológico grant 1100298. J. Gärtner, R. Sun, and partially A.F. Ramírez were supported by the DFG Forschergruppe 718 Analysis and Stochastics in Complex Physical Systems.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alexander Drewitz
    • 1
  • Jürgen Gärtner
    • 2
  • Alejandro F. Ramírez
    • 3
  • Rongfeng Sun
    • 4
  1. 1.Departement MathematikEidgenössische Technische Hochschule ZürichZürichSwitzerland
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  4. 4.Department of MathematicsNational University of SingaporeSingaporeSingapore

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