A Remark on Localization for Branching Random Walks in Random Environment

Hadrian Heil (Universität Tübingen)
Makoto Nakashima (Kyoto University)

Abstract


We prove a localization-result for branching random walks in random environment, namely that if the process does not die out, the most populated site will infinitely often contain more than a fixed percentage of the population. This had been proven already before by Hu and Yoshida, but it is possible to drop their assumption that particles may not die.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 323-336

Publication Date: June 22, 2011

DOI: 10.1214/ECP.v16-1603

References

  1. Krishna B. Athreya, Peter E. Ney. Branching processes. Die Grundlehren der mathematischen Wissenschaften, Band 196. Springer-Verlag, New York-Heidelberg, 1972. MR0373040
  2. Matthias Birkner, Jochen Geiger, Götz Kersting. Branching processes in random environment---a view on critical and subcritical cases. Interacting stochastic systems, 269--291, Springer, Berlin, 2005. MR2118578
  3. Matthias Birkner. Particle Systems with locally dependent branching: Long-time-behaviour, genealogy and critical parameters. Ph.D. thesis, Johann Wolfgang Goethe-Universität, Frankfurt.
  4. Francis Comets and Nobuo Yoshida. Branching Random Walks in Space-Time Random Environment: Survival Probability, Global and Local Growth Rates. J. Theor. Probab., to appear.
  5. Durrett, Richard. Probability. Theory and examples. The Wadsworth \& Brooks/Cole Statistics/Probability Series. Wadsworth \& Brooks/Cole Advanced Books \& Software, Pacific Grove, CA, 1991. MR1068527
  6. Hadrian Heil, Makoto Nakashima, Nobuo Yoshida. Branching Random Walks in Random Environment in d>3 are Diffusive in the Regular Growth Phase. Electron. J. Prob., to appear, 2011.
  7. Yueyun Hu, Nobuo Yoshida. Localization for branching random walks in random environment. Stochastic Process. Appl. 119 (2009), no. 5, 1632--1651. MR2513122
  8. Makoto Nakashima. Almost Sure Central Limit theorem for Branching Random Walks in Random Environment. Ann. Appl. Probab., to appear, 2010.
  9. Nobuo Yoshida. Localization for linear stochastic evolutions. J. Stat. Phys. 138 (2010), no. 4-5, 598--618. MR2594914
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.