  In Phys. Rev. A {\bf 71}, 060312(R) (2005) the robustness of the local
adiabatic quantum search to decoherence in the instantaneous eigenbasis of the
search Hamiltonian was examined. We expand this analysis to include the case of
the global adiabatic quantum search. As in the case of the local search the
asymptotic time complexity for the global search is the same as for the ideal
closed case, as long as the Hamiltonian dynamics is present. In the case of
pure decoherence, where the environment monitors the search Hamiltonian, we
find that the time complexity of the global quantum adiabatic search scales
like $N^{3/2}$, where $N$ is the list length. We moreover extend the analysis
to include success probabilities $p<1$ and prove bounds on the run time with
the same scaling as in the conditions for the $p\to 1$ limit. We supplement the
analytical results by numerical simulations of the global and local search.
