  In order to generate more than 60 e-folds of accelerated expansion in
original hybrid inflation, 2-fields trajectories are usually required to be
initially fine-tuned in a very narrow band along the inflationary valley or in
some isolated points outside it. From a more precise investigation of the
dynamics, these points which can cover a non-negligible proportion of the space
of sub-planckian initial field values, depending on the potential parameters,
are shown to be organised in connected domains with fractal boundaries. They
correspond to trajectories first falling towards the bottom of the potential,
then climbing and slow-rolling back along the inflationary valley. The full
parameter space, including initial velocities and all the potential parameters,
is then explored by using Monte-Carlo-Markov-Chains (MCMC) methods. Results
indicate that successful initial conditions (IC) outside the valley are not
localized in the parameter space and are the dominant way to realise inflation,
independently of initial field velocities. Natural bounds on parameters are
deduced. The genericity of our results is confirmed in 5 other hybrid models
from various framework.
