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Annealed survival asymptotics for Brownian motion in a scaled Poissonian potential

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  • Merkl, Franz
  • Wüthrich, Mario V.

Abstract

We consider d-dimensional Brownian motion evolving in a scaled Poissonian potential [beta][phi]-2(t)V, where [beta]>0 is a constant, [phi] is the scaling function which typically tends to infinity, and V is obtained by translating a fixed non-negative compactly supported shape function to all the particles of a d-dimensional Poissonian point process. We are interested in the large t behavior of the annealed partition sum of Brownian motion up to time t under the influence of the natural Feynman-Kac weight associated to [beta][phi]-2(t)V. We prove that for d[greater-or-equal, slanted]2 there is a critical scale [phi] and a critical constant [beta]c(d)>0 such that the annealed partition sum undergoes a phase transition if [beta] crosses [beta]c(d). In d=1 this picture does not hold true, which can formally be interpreted that on the critical scale [phi] we have [beta]c(1)=0.

Suggested Citation

  • Merkl, Franz & Wüthrich, Mario V., 2001. "Annealed survival asymptotics for Brownian motion in a scaled Poissonian potential," Stochastic Processes and their Applications, Elsevier, vol. 96(2), pages 191-211, December.
    • Handle: RePEc:eee:spapps:v:96:y:2001:i:2:p:191-211
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    Cited by:

    1. Sue, Shih-Che & Chang, Chi-Fon & Huang, Yao-Te & Chou, Ching-Yu & Huang, Tai-huang, 2005. "Challenges in NMR-based structural genomics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 350(1), pages 12-27.

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