Branching random walks in time inhomogeneous environments

Ming Fang (University of Minnesota)
Ofer Zeitouni (University of Minnesota; Weizmann Institute.)

Abstract


We study the maximal displacement of branching random walks in a class of time inhomogeneous environments. Specifically, binary branching random walks with Gaussian increments will be considered, where the variances of the increments change over time macroscopically. We find the asymptotics of the maximum up to an $O_P(1)$ (stochastically bounded) error, and focus on the following phenomena: the profile of the variance matters, both to the leading (velocity) term and to the logarithmic correction term, and the latter exhibits a phase transition.

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

Pages: 1-18

Publication Date: August 19, 2012

DOI: 10.1214/EJP.v17-2253

References

  • Addario-Berry, Louigi; Reed, Bruce. Minima in branching random walks. Ann. Probab. 37 (2009), no. 3, 1044--1079. MR2537549
  • E. Aïdékon: Convergence in law of the minimum of a branching random walk, ARXIV1101.1810v3
  • Alon, Noga; Spencer, Joel H. The probabilistic method. Third edition. With an appendix on the life and work of Paul Erdős. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., Hoboken, NJ, 2008. xviii+352 pp. ISBN: 978-0-470-17020-5 MR2437651
  • Berestycki, J.; Brunet, É.; Harris, J. W.; Harris, S. C. The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential. Statist. Probab. Lett. 80 (2010), no. 17-18, 1442--1446. MR2669786
  • J. Berestycki, É. Brunet, J. W. Harris, S. C. Harris, and M. I. Roberts: Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential, ARXIV1203.0513
  • Bramson, Maury D. Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 (1978), no. 5, 531--581. MR0494541
  • Bramson, Maury D. Minimal displacement of branching random walk. Z. Wahrsch. Verw. Gebiete 45 (1978), no. 2, 89--108. MR0510529
  • Bramson, Maury; Zeitouni, Ofer. Tightness for a family of recursion equations. Ann. Probab. 37 (2009), no. 2, 615--653. MR2510018
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Second edition. Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2 MR1619036
  • L. Doering and M.I. Roberts: Catalytic branching processes via spine techniques and renewal theory, ARXIV1106.5428v4
  • Durrett, Rick. Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8 MR2722836
  • Engländer, János; Harris, Simon C.; Kyprianou, Andreas E. Strong law of large numbers for branching diffusions. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 1, 279--298. MR2641779
  • M. Fang: Tightness for maxima of generalized branching random walks. phJournal of Applied Probability, 49(3), (2012).
  • M. Fang and O. Zeitouni: Slowdown for time inhomogeneous branching brownian motion, ARXIV1205.1769
  • Gantert, Nina; Müller, Sebastian; Popov, Serguei; Vachkovskaia, Marina. Survival of branching random walks in random environment. J. Theoret. Probab. 23 (2010), no. 4, 1002--1014. MR2735734
  • Git, Y.; Harris, J. W.; Harris, S. C. Exponential growth rates in a typed branching diffusion. Ann. Appl. Probab. 17 (2007), no. 2, 609--653. MR2308337
  • Greven, Andreas; den Hollander, Frank. Branching random walk in random environment: phase transitions for local and global growth rates. Probab. Theory Related Fields 91 (1992), no. 2, 195--249. MR1147615
  • Harris, J. W.; Harris, S. C. Branching Brownian motion with an inhomogeneous breeding potential. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 3, 793--801. MR2548504
  • S. C. Harris and M. I. Roberts: The many-to-few lemma and multiple spines, ARXIV1106.4761
  • Harris, S. C.; Williams, D. Large deviations and martingales for a typed branching diffusion. I. Hommage à P. A. Meyer et J. Neveu. Astérisque No. 236 (1996), 133--154. MR1417979
  • Heil, Hadrian; Nakashima, Makoto; Yoshida, Nobuo. Branching random walks in random environment are diffusive in the regular growth phase. Electron. J. Probab. 16 (2011), no. 48, 1316--1340. MR2827461
  • Hu, Yueyun; Yoshida, Nobuo. Localization for branching random walks in random environment. Stochastic Process. Appl. 119 (2009), no. 5, 1632--1651. MR2513122
  • L. Koralov: Branching diffusion in inhomogeneous media, ARXIV1107.1159
  • Lau, Ka-Sing. On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov. J. Differential Equations 59 (1985), no. 1, 44--70. MR0803086
  • Q. Liu: Branching random walks in random environment. phICCM 2, (2007), 702--719.
  • Machado, F. P.; Popov, S. Yu. One-dimensional branching random walks in a Markovian random environment. J. Appl. Probab. 37 (2000), no. 4, 1157--1163. MR1808881
  • Nakashima, Makoto. Almost sure central limit theorem for branching random walks in random environment. Ann. Appl. Probab. 21 (2011), no. 1, 351--373. MR2759206
  • Nolen, James; Ryzhik, Lenya. Traveling waves in a one-dimensional heterogeneous medium. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 3, 1021--1047. MR2526414
  • M. I. Roberts: A simple path to asymptotics for the frontier of a branching Brownian motion, to appear in Annals Probab., ARXIV1106.4771
Bookmark and Share


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