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Brownian survival among Poissonian traps with random shapes at critical intensity
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  • Published: 10 February 2005

Brownian survival among Poissonian traps with random shapes at critical intensity

  • M. van den Berg1,
  • E. Bolthausen2 &
  • F. den Hollander3 

Probability Theory and Related Fields volume 132, pages 163–202 (2005)Cite this article

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Abstract.

In this paper we consider a standard Brownian motion in ℝd, starting at 0 and observed until time t. The Brownian motion takes place in the presence of a Poisson random field of traps, whose centers have intensity ν t and whose shapes are drawn randomly and independently according to a probability distribution Π, on the set of closed subsets of ℝd, subject to appropriate conditions. The Brownian motion is killed as soon as it hits one of the traps. With the help of a large deviation technique developed in an earlier paper, we find the tail of the probability S t that the Brownian motion survives up to time t when

where c ∈ (0,∞) is a parameter. This choice of intensity corresponds to a critical scaling. We give a detailed analysis of the rate constant in the tail of S t as a function of c, including its limiting behaviour as c→∞ or c↓0. For d≥3, we find that there are two regimes, depending on the choice of Π. In one of the regimes there is a collapse transition at a critical value c* ∈ (0,∞), where the optimal survival strategy changes from being diffusive to being subdiffusive. At c*, the slope of the rate constant is discontinuous. For d=2, there is again a collapse transition, but the rate constant is independent of Π and its slope at c=c* is continuous.

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom

    M. van den Berg

  2. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland

    E. Bolthausen

  3. EURANDOM, P.O. Box 513, 5600, MB, Eindhoven, The Netherlands

    F. den Hollander

Authors
  1. M. van den Berg
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  2. E. Bolthausen
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  3. F. den Hollander
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Corresponding author

Correspondence to M. van den Berg.

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Mathematics Subject Classification (2000): 60F10, 60G50, 35J20

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den Berg, M., Bolthausen, E. & den Hollander, F. Brownian survival among Poissonian traps with random shapes at critical intensity. Probab. Theory Relat. Fields 132, 163–202 (2005). https://doi.org/10.1007/s00440-004-0393-4

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  • Received: 09 December 2003

  • Revised: 09 September 2004

  • Published: 10 February 2005

  • Issue Date: June 2005

  • DOI: https://doi.org/10.1007/s00440-004-0393-4

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Keywords

  • Brownian motion
  • Poisson random field of traps
  • Random capacity
  • Survival probability
  • Large deviations
  • Variational problems
  • Sobolev inequalities
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