International Journal of Stochastic Analysis
Volume 2011 (2011), Article ID 605068, 37 pages
Research Article

Nonconservative Diffusions on [ 0 , 1 ] with Killing and Branching: Applications to Wright-Fisher Models with or without Selection

Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8089 et Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France

Received 22 November 2010; Accepted 9 May 2011

Academic Editor: Manuel O. Cáceres

Copyright © 2011 Thierry E. Huillet. 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 consider nonconservative diffusion processes 𝑥 𝑡 on the unit interval, so with absorbing barriers. Using Doob-transformation techniques involving superharmonic functions, we modify the original process to form a new diffusion process 𝑥 𝑡 presenting an additional killing rate part 𝑑 > 0 . We limit ourselves to situations for which 𝑥 𝑡 is itself nonconservative with upper bounded killing rate. For this transformed process, we study various conditionings on events pertaining to both the killing and the absorption times. We introduce the idea of a reciprocal Doob transform: we start from the process 𝑥 𝑡 , apply the reciprocal Doob transform ending up in a new process which is 𝑥 𝑡 but now with an additional branching rate 𝑏 > 0 , which is also upper bounded. For this supercritical binary branching diffusion, there is a tradeoff between branching events giving birth to new particles and absorption at the boundaries, killing the particles. Under our assumptions, the branching diffusion process gets eventually globally extinct in finite time. We apply these ideas to diffusion processes arising in population genetics. In this setup, the process 𝑥 𝑡 is a Wright-Fisher diffusion with selection. Using an exponential Doob transform, we end up with a killed neutral Wright-Fisher diffusion 𝑥 𝑡 . We give a detailed study of the binary branching diffusion process obtained by using the corresponding reciprocal Doob transform.