Nonconservative Diffusions on [0,1] with Killing and Branching: Applications to Wright-Fisher Models with or without Selection (original) (raw)
2011, International Journal of Stochastic Analysis
We consider nonconservative diffusion processes 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 presenting an additional killing rate part . We limit ourselves to situations for which 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 , apply the reciprocal Doob transform ending up in a new process which is but now with an additional branching rate , 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...
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