  (Abridged) We study relationships between the SFR and the nuclear properties
of X-ray selected AGNs out to z=2.5, using far-IR data in three extragalactic
deep fields as part of the PACS Evolutionary Probe (PEP) program. Guided by
studies of intrinsic infra-red AGN SEDs, we show that the majority of the FIR
emission in AGNs is produced by cold dust heated by star-formation. We uncover
characteristic redshift-dependent trends between the mean FIR luminosity
(L_fir) and accretion luminosity (L_agn) of AGNs. At low AGN luminosities,
accretion and SFR are uncorrelated at all redshifts, consistent with a scenario
where most low-luminosity AGNs are primarily fueled by secular processes in
their host galaxies. At high AGN luminosities, a significant correlation is
observed between L_fir and L_agn, but only among AGNs at low and moderate
redshifts (z<1). We interpret this as a signature of the increasing importance
of major-mergers in driving both the growth of super-massive black holes
(SMBHs) and global star-formation in their hosts at high AGN luminosities.
However, we also find that the enhancement of SFR in luminous AGNs weakens or
disappears at high redshifts (z>1). This suggests that the role of mergers in
SMBH-galaxy co-evolution is less important at these epochs. At all redshifts,
we find essentially no relationship between L_fir and nuclear obscuration
across five orders of magnitude in obscuring column density, suggesting that
various different mechanisms are likely to be responsible for obscuring X-rays
in active galaxies. We explain our results within a scenario in which two
different modes of SMBH fueling operate among low- and high-luminosity AGNs. We
postulate, guided by emerging knowledge about the properties of high redshift
galaxies, that the dominant mode of accretion among high-luminosity AGNs
evolves with redshift.
