  The formation, merging, and accretion history of massive black holes along
the hierarchical build--up of cosmic structures leaves a unique imprint on the
background of gravitational waves at mHz frequencies. We study here, by means
of dedicated simulations of black hole build--up, the possibility of
constraining different models of black hole cosmic evolution using future
gravitational wave space--borne missions, such as LISA. We consider two main
scenarios for black hole formation, namely, one where seeds are light (~10^2
\msun, remnant of Population III stars), and one where seeds are heavy (>~10^4
\msun, direct collapse). In all the models we have investigated, massive black
hole binary coalescences do not produce a stochastic GW background, but rather,
a set of individual resolved events. Detection of several hundreds merging
events in 3 year LISA mission will be the sign of a heavy seed scenario with
efficient formation of black hole seeds in a large fraction of high redshift
halos. On the other extreme, a low event rate, about few tens in 3 years, is
peculiar of scenarios where either the seeds are light, and many coalescences
do not fall into the LISA band, or seeds are massive, but rare. In this case a
decisive diagnostic is provided by the shape of the mass distribution of
detected events. Light binaries (m<10^4\msun) are predicted in a fairly large
number in Population III remnants models, but are totally absent in direct
collapse models. Finally, a further, helpful diagnostic of black hole formation
models lies in the distribution of the mass ratios in binary coalescences.
While heavy seed models predict that most of the detected events involve equal
mass binaries, in the case of light seeds, mass ratios are equally distributed
in the range 0.1-1.
