  Inward migration of giant planets is predicted by hydrodynamical simulations
during the gas phase of the protoplanetary disc. The phenomenon is also invoked
to explain resonant and near-resonant exoplanetary system structures. The early
inward migration may also have affected our Solar System and sculpted its
different minor planet reservoirs. In this study we explore how the early
inward migration of the giant planets shapes the Kuiper Belt. We test different
scenarios with only Neptune and Uranus and with all the four giant planets,
including also some models with the subsequent outward planetesimal-driven
migration of Neptune after the gas dispersal. We find objects populating mean
motion resonances even when Neptune and Uranus do not migrate at all or only
migrate inwards. When the planets are fixed, planetesimals stick only
temporarily to the mean motion resonances, while inwards migration yields a new
channel to populate the resonances without invoking convergent migration. In
these cases, however, it is hard to populate mean motion resonances that do not
cross the planetesimal disc (such as 2:1 and 5:2) and there is a lack of
resonant KBOs that cross Neptune's orbit. These Neptune crossers are an
unambiguous signature of the outward migration of Neptune. The starting
position and the growth rate of Neptune matters for the contamination of the
classical Kuiper belt region from neighbouring regions. The eccentricity and
inclination space of the hot classicals and the scattered disc region become
much more populated when all the giant planets are included. The 5:2 resonance
with Neptune becomes increasingly populated with deeper inward migrations of
Neptune. The overall inclination distribution, however, is still narrower than
from observations, as is generally the case for Kuiper belt population models.
