  We derive a theoretical model which describes Bose-Einstein condensation in
an open driven-dissipative system. It includes external pumping of a thermal
reservoir, finite life time of the condensed particles and energy relaxation.
The coupling between the reservoir and the condensate is described with
semi-classical Boltzmann rates. This results in a dissipative term in the
Gross-Pitaevskii equation for the condensate, which is proportional to the
energy of the elementary excitations of the system. We analyse the main
properties of a condensate described by this hybrid Boltzmann Gross-Pitaevskii
model, namely, dispersion of the elementary excitations, bogolon distribution
function, first order coherence, dynamic and energetic stability, drag force
created by a disorder potential. We find that the dispersion of the elementary
excitations of a condensed state fulfils the Landau criterion of superfluidity.
The condensate is dynamically and energetically stable as longs it moves at a
velocity smaller than the speed of excitations. First order spatial coherence
of the condensate is found to decay exponentially in 1D and with a power law in
2D, similarly with the case of conservative systems. The coherence lengths are
found to be longer due to the finite life time of the condensate excitations.
We compare these properties with the ones of a condensate described by the
popular "diffusive" models in which the dissipative term is proportional to the
local condensate density. In the latter, the dispersion of excitations is
diffusive which as soon as the condensate is put into motion implies finite
mechanical friction and can lead to an energetic instability.
