  Planetesimal formation is still mysterious. One of the ways to form
planetesimals is to invoke a gas pressure bump in a protoplanetary disc. In our
previous paper, we propose a new scenario in which the piled-up dust at a gas
pressure bump created by a migrating planet form planetesimals by streaming
instability in a wide region of the disc as the planet migrates inward. In this
work, we consider the global time evolution of dust and investigate the
detailed conditions and results of the planetesimal formation in our scenario.
We use a 1D grid single-sized dust evolution model, which can follow the growth
of the particles by their mutual collision and their radial drift and
diffusion. We calculate the time-evolution of the radial distribution of the
peak mass and surface density of the dust in a gas disc perturbed by an
embedded migrating planet and investigate if the dust satisfies the condition
for planetesimal formation. We find that planetesimals form in a belt-like
region between the snowline and the position where the planet reaches its
pebble-isolation mass when the strength of turbulence is
$10^{-4}\leq\alpha\leq10^{-3}$, which is broadly consistent with observed
value. The mechanism of the formation, streaming instability or mutual
collision, depends on the timescale of the streaming instability. The total
mass of planetesimals also depends on $\alpha$ and is about $30-100~M_{\rm E}$
if the planetary core has already existed at the beginning and grows by gas
accretion, but it decreases as the timing of the formation of the planetary
core is later. We also provide simple approximate expressions of the surface
density and total mass of the planetesimals and find that the total mass
strongly depends on the dust mass. We show that planetesimals form in a
belt-like region by the combination of the dust pile-up at the gas pressure
bump formed by a planet and its inward migration.
