A generalized Wigner–Moyal statistical theory of radiation is used to obtain a general dispersion relation for stimulated Brillouin Scattering (SBS) driven by a broadband radiation field with arbitrary statistics. The monochromatic limit is recovered from our general result, reproducing the classic monochromatic dispersion relation. The behavior of the growth rate of the instability as a simultaneous function of the bandwidth of the pump wave, the intensity of the incident field and the wave number of the scattered wave is further explored by numerically solving the dispersion relation. Our results show that the growth rate of SBS can be reduced by 1/3 for a bandwidth of 0.3 nm, for typical experimental parameters.