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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
van den Berg, M., Bolthausen, E.: Asymptotics of the generating function for the volume of the Wiener sausage. Probab. Theory Relat. Fields 99, 389–397 (1994)
van den Berg, M., Bolthausen, E., den Hollander, F.: Moderate deviations for the volume of the Wiener sausage. Ann. Math. 153, 355–406 (2001)
van den Berg, M., Bolthausen, E.,den Hollander, F.: On the volume of the intersection of two Wiener sausages. Ann. Math. 159, 741–782 (2004)
Bryc, W.: Large deviations by the asymptotic value method. In: Diffusion Processes and Related Problems in Analysis. M. Pinsky (ed.), Vol. 1, pp. 447–472, Birkhäuser, Boston, 1990
Le Gall, J.-F.: Sur la saucisse de Wiener et les points multiples du mouvement brownien. Ann. Probab. 14, 1219–1244 (1986)
Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics 14, AMS, Providence RI, 1997
Merkl, F., Wüthrich, M.V.: Phase transition of the principal Dirichlet eigenvalue in a scaled Poissonian potential. Probab. Theory Relat. Fields 119, 475–507 (2001)
Merkl, F., Wüthrich, M.V.: Annealed survival asymptotics for Brownian motion in a scaled Poissonian potential. Stoch. Proc. Appl. 96, 191–211 (2001)
Merkl, F., Wüthrich, M.V.: Infinite volume asymptotics of the ground state energy in a scaled Poissonian potential. Ann. Inst. H. Poincaré Probab. Statist. 38, 253–284 (2002)
Spitzer, F.: Electrostatic capacity, heat flow and Brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb. 3, 110–121 (1964)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): 60F10, 60G50, 35J20
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-004-0393-4