  We consider diffusion processes x_{t} on the unit interval.
Doob-transformation techniques consist of a selection of x_{t}-paths procedure.
The law of the transformed process is the one of a branching diffusion system
of particles, each diffusing like a new process tilde{x}_{t}, superposing an
additional drift to the one of x_{t}. Killing and/or branching of
tilde{x}_{t}-particles occur at some space-dependent rate lambda. For this
transformed process, so in the class of branching diffusions, the question
arises as to whether the particle system is sub-critical, critical or
super-critical. In the first two cases, extinction occurs with probability one.
We apply this circle of ideas to diffusion processes arising in population
genetics. In this setup, the process x_{t} is a Wright-Fisher (WF) diffusion,
either neutral or with mutations. We study a particular Doob transform which is
based on the exponential function in the usual fitness parameter sigma. We have
in mind that this is an alternative way to introduce selection or fitness in
both WF-like diffusions, leading to branching diffusion models ideas. For this
Doob-transform model of fitness, the usual selection drift sigma x(1-x) should
be superposed to the one of x_{t} to form tilde{x}_{t} which is the process
that can branch, binarily. In the first neutral case, there is a trade-off
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 with
exponential tails. In the second case with mutations, there is a trade-off
between killing events removing some particles from the system and reflection
at the boundaries where the particles survive. This branching diffusion process
also gets eventually globally extinct but in very long finite time with
power-law tails. Our approach relies on the spectral expansion of the
transition probability kernels of both x_{t} and tilde{x}_{t}.
