  Assuming that primordial black holes compose a fraction of dark matter, some
of them may accumulate at the center of galaxy and perform a prograde or
retrograde orbit against the gravity pointing towards the center exerted by the
central massive black hole. If the mass of primordial black holes is of the
order of stellar mass or smaller, such extreme mass ratio inspirals can emit
gravitational waves and form a background due to incoherent superposition of
all the contributions of the Universe. We investigate the stochastic
gravitational-wave background energy density spectra from the directional
source, the primordial black holes surrounding Sagittarius A$^\ast$ of the
Milky Way, and the isotropic extragalactic total contribution, respectively. As
will be shown, the resultant stochastic gravitational-wave background energy
density shows different spectrum features such as the peak positions in the
frequency domain for the above two kinds of sources. Detection of stochastic
gravitational-wave background with such a feature may provide evidence for the
existence of primordial black holes. Conversely, a null searching result can
put constraints on the abundance of primordial black holes in dark matter.
