[Abridged] We investigate the global galaxy evolution over 12 Gyr (0.05<z<4.5), from the star formation rate density (SFRD), combining the VVDS Deep (17.5<=I<=24.0) and Ultra-Deep (23.00<=i<=24.75) surveys. We obtain a single homogeneous spectroscopic redshift sample, totalizing about 11000 galaxies. We estimate the rest-frame FUV luminosity function (LF) and luminosity density (LD), extract the dust attenuation of the FUV radiation using SED fitting, and derive the dust-corrected SFRD. We find a constant and flat faint-end slope alpha in the FUV LF at z<1.7. At z>1.7, we set alpha steepening with (1+z). The absolute magnitude M*_FUV brightens in the entire range 0<z<4.5, and at z>2 it is on average brighter than in the literature, while phi* is smaller. Our total LD shows a peak at z=2, present also when considering all sources of uncertainty. The SFRD history peaks as well at z=2. It rises by a factor of 6 during 2 Gyr (from z=4.5 to z=2), and then decreases by a factor of 12 during 10 Gyr down to z=0.05. This peak is mainly produced by a similar peak within the population of galaxies with -21.5<=M_FUV<=-19.5 mag. As times goes by, the total SFRD is dominated by fainter and fainter galaxies. The presence of a clear peak at z=2 and a fast rise at z>2 of the SFRD is compelling for models of galaxy formation. The mean dust attenuation A_FUV of the global galaxy population rises by 1 mag during 2 Gyr from z=4.5 to z=2, reaches its maximum at z=1 (A_FUV=2.2 mag), and then decreases by 1.1 mag during 7 Gyr down to z=0. The dust attenuation maximum is reached 2 Gyr after the SFRD peak, implying a contribution from the intermediate-mass stars to the dust production at z<2.
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