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

We consider nonconservative diffusion processes 𝑥𝑡 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 𝑥𝑡 presenting an additional killing rate part 𝑑>0. We limit ourselves to situations for which 𝑥𝑡 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 𝑥𝑡, apply the reciprocal Doob transform ending up in a new process which is 𝑥𝑡 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 𝑥𝑡 is a Wright-Fisher diffusion with selection. Using an exponential Doob transform, we end up with a killed neutral Wright-Fisher diffusion 𝑥𝑡. We give a detailed study of the binary branching diffusion process obtained by using the corresponding reciprocal Doob transform.