Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 60376, 23 pages

Likely path to extinction in simple branching models with large initial population

F. C. Klebaner and R. Liptser

School of Mathematical Sciences, Monash University, University of Sciences and Technology Houary Boumediene, Victoria 3800, Australia

Received 15 September 2005; Revised 24 November 2005; Accepted 4 December 2005

Copyright © 2006 F. C. Klebaner and R. Liptser. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We give explicit formulae for most likely paths to extinction in simple branching models when initial population is large. In discrete time, we study the Galton-Watson process, and in continuous time, the branching diffusion. The most likely paths are found with the help of the large deviation principle (LDP). We also find asymptotics for the extinction probability, which gives a new expression in continuous time and recovers the known formula in discrete time. Due to the nonnegativity of the processes, the proof of LDP at the point of extinction uses a nonstandard argument of independent interest.