On-off intermittency is an aperiodic switching between static, or lami
nar, behavior and bursts of oscillations. It has been recently investi
gated in a class of one-dimensional maps that are multiplicatively cou
pled to either random or chaotic signals. The addition of noise to the
system results in a fundamental change in the nature of the intermitt
ency. The critical onset parameter changes discontinuously, and the di
stribution of the laminar phases is modified. These contrasts are exam
ined in terms of random walks.