  We compute the power spectrum of the noise in the current due to spin motive
forces by a fluctuating domain wall. We find that the power spectrum of the
noise in the current is colored, and depends on the Gilbert damping, the spin
transfer torque parameter $\beta$, and the domain-wall pinning potential and
magnetic anisotropy. We also determine the average current induced by the
thermally-assisted motion of a domain wall that is driven by an external
magnetic field. Our results suggest that measuring the power spectrum of the
noise in the current in the presence of a domain wall may provide a new method
for characterizing the current-to-domain-wall coupling in the system.
