  We use theoretical analysis and numerical simulation to investigate the
operation of a laser oscillating from gain supplied by stimulated Brillouin
scattering (SBS) in a microresonator. The interaction of the forward, backward,
and density waves within the microresonator results in a set of coupled-mode
equations describing both the laser's phase and amplitude evolution over time.
Using this coupled-mode formalism, we investigate the performance of the SBS
laser under noise perturbation and identify the fundamental parameters and
their optimization to enable low-noise SBS operation. The intrinsic laser
linewidth, which is primarily limited by incoherent thermal occupation of the
density wave, can be of order hertz or below. Our analysis also determines the
SBS laser's relaxation oscillation, which results from the coupling between the
optical and density waves, and appears as a resonance in both the phase and
amplitude quadratures. We further explore contributions of the pump noise to
the SBS laser's performance, which we find under most circumstances to increase
the SBS laser noise beyond its fundamental limits. By tightly stabilizing the
pump laser onto the microcavity resonance, the transfer of pump noise is
significantly reduced. Our analysis is both supported and extended through
numerical simulations of the SBS laser.
