  We study the effects of noise on the collective dynamics of an ensemble of
coupled phase oscillators whose natural frequencies are all identical, but
whose coupling strengths are not the same all over the ensemble. The intensity
of noise can also be heterogeneous, representing diversity in the individual
responses to external fluctuations. We show that the desynchronization
transition induced by noise may be completely suppressed, even for arbitrarily
large noise intensities, is the distribution of coupling strengths decays
slowly enough for large couplings. Equivalently, if the response to noise of a
sufficiently large fraction of the ensemble is weak enough, desynchronization
cannot occur. The two effects combine with each other when the response to
noise and the coupling strength of each oscillator are correlated. This
combination is quantitatively characterized and illustrated with explicit
examples.
