Summary
A particle is considered which moves in ℝd according to a Brownian motion with drifth≠0. The space is assumed to contain random traps. The probability of survival of the particle up to timeT decays exponentially asT→∞ with a positive decay rate λ. λ is shown to be a non-analytic function of |h|. For small |h| the decay rate is given by λ(h)=1/2|h|2; but if |h| exceeds a certain critical value, λ(h) depends also on the parameters describing trapping. Upper and lower bounds for λ(h) are given, which imply the asymptotic linearity of λ(h) for large |h|. The critical point marks a transition from localized to delocalized behavior. A variational formula for the decay rate is given on the level of generalized processes, which elucidates the mathematical mechanism behind observations made earlier by Grassberger and Procaccia on the basis of computer simulations.
References
Donsker, M.D., Varadhan, S.R.S.: Asymptotics for the Wiener sausage. Comm. Pure Appl. Math.28, 525–565 (1975)
Grassberger, P., Procaccia, I.: Diffusion and drift in a medium with randomly distributed traps. Phys. Rev. A26, 3686–3688 (1982)
Hida, T.: Brownian motion. Berlin Heidelberg New York: Springer 1980
Kac, M.: Probability and related topics in physical sciences. London: Interscience-Wiley 1959
Kang, K., Redner, S.: Novel behavior of biased correlated walks in one dimension. J. Chem. Phys.80, 2752–2755 (1984)
Kusuoka, S.: The variational principle for stationary Gaussian Markov fields. In: Kallianpur, G. (ed.) Lect. Notes Control Inform.49, 179–187 (1983)
Kusuoka, S.: Asymptotics of polymer measures in one dimension. In: Proc. Bielefeld meeting “Stochastic Processes and Infinite Dimensional Analysis” 1984
Movaghar, B., Pohlmann, B., Wurtz, D.: Electric field dependence of tropping in one dimension. Phys. Rev. A29, 1568–1570 (1984)
Spitzer, F.: Electrostatic capacity, heat flow, and Brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 110–121 (1964)
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Supported by Deutsche Forschungsgemeinschaft
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Eisele, T., Lang, R. Asymptotics for the wiener sausage with drift. Probab. Th. Rel. Fields 74, 125–140 (1987). https://doi.org/10.1007/BF01845643
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DOI: https://doi.org/10.1007/BF01845643
Keywords
- Computer Simulation
- Lower Bound
- Stochastic Process
- Brownian Motion
- Decay Rate