Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T02:32:43.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Slow variation and uniqueness of solutions to the functional equation in the branching random walk

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

A. E. Kyprianou
A. E. Kyprianou*
Affiliation:
The London School of Economics
*
∗Postal address: Department of Mathematics and Statistics, University of Edinburgh, James Clerk Maxwell Building, Mayfield Road, Edinburgh, EH9 3JZ, Scotland. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this short communication, some of the recent results of Liu (1998) and Biggins and Kyprianou (1997), concerning solutions to a certain functional equation associated with the branching random walk, are strengthened. Their importance is emphasized in the context of travelling wave solutions to a discrete version of the KPP equation and the connection with the behaviour of the rightmost particle in the nth generation.

Keywords

Branching random walkfunctional equationsmartingalesKPP equationtravelling wave solutions

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 795 - 801
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baringhaus, L. and Grübel, R. (1995). Random convex combinations of a new characterization of exponential distributions. Institut für Mathematische Stochastik, Universität Hannover.Google Scholar
Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.Google Scholar
Biggins, J. D. (1991). Uniform convergence of martingales in the one dimensional branching random walk. In Selected Proceedings of Sheffield Symposium on Applied Probability, ed. Baswa, I. V. and Taylor, R. L., IMS Lecture Notes Monograph Series 18, 159–173.Google Scholar
Biggins, J. D., and Kyprianou, A. E. (1997). Seneta–Heyde norming in the branching random walk. Ann. Prob. 25, 337–360.CrossRefGoogle Scholar
Bramson, M. (1978). Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. XXXI, 531–581.Google Scholar
Chauvin, B., and Roualt, A. (1997). Boltzmann–Gibbs weights in the branching random walk. In Classical and Modern Branching Processes, ed. Athreya, K. B. and Jagers, P., IMA Proceedings 84, 41–50.Google Scholar
Cohn, H. (1985). A martingale approach to supercritical (CMJ) branching processes. Ann. Prob. 13, 1179–1191.Google Scholar
Durrett, R., and Liggett, M. (1983). Fixed points of the smoothing transform. Z. Wahrscheinlichkeitsth. 64, 275–301.CrossRefGoogle Scholar
Kahane, J. P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, 131–145.CrossRefGoogle Scholar
Kingman, J. F. C. (1975). The first birth problem for an age-dependent branching process. Ann. Prob. 3, 790–801.Google Scholar
Kolmogorov, A. N., and Fomin, S. V. (1970). Introductory Real Analysis. Dover, New York.Google Scholar
Kyprianou, A. E. (1996). Seneta–Heyde norming in spatial branching processes and associated problems. , University of Sheffield.Google Scholar
Lalley, S. P., and Sellke, T. (1987). A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Prob. 15, 1052–1061.Google Scholar
Liu, Q. (1997). Sur une équation fonctionelle et ses applications: une extension du théorème de Kesten–Stigum concernant des processus de branchement. Adv. Appl. Prob. 29, 353–373.CrossRefGoogle Scholar
Liu, Q. (1998). Fixed points of a generalized smoothing transformation and its applications to the branching random walk. Adv. Appl. Prob. 30, 85–113.Google Scholar
McDiarmid, C. (1995). Minimal positions in a branching random walk. Ann. Appl. Prob. 5, 128–139.Google Scholar
McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Comm. Pure Appl. Math. XXIX, 553–554.Google Scholar
Neveu, J. (1988). Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes 1987, ed. Çinlar, E., Chung, K. L. and Getoor, R. K. (Progress in Probability and Statistics 15). Birkhaüser, Boston, pp. 223–241.Google Scholar
Pakes, A. G. (1992). On characterizations through mixed sums. Austral. J. Statist. 34, 323–339.Google Scholar
Schuh, H. J. (1982). Seneta constants for the supercritical Bellman–Harris process. Adv. Appl. Prob. 14, 732–751.CrossRefGoogle Scholar
Waymire, E., and Williams, S. (1996). A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Amer. Math. Soc. 348, 585–632.Google Scholar