  Gas accretion of embedded stellar-mass black holes\,(sBHs) or stars in the
accretion disk of active galactic nuclei\,(AGNs) will modify the mass
distribution of these sBHs and stars, which will also affect the migration of
the sBHs/stars. With the introduction of the mass accretion effect, we simulate
the evolution of the sBH/star distribution function in a consistent way by
extending the Fokker-Planck equation of sBH/star distributions to the
mass-varying scenario, and explore the mass distribution of sBHs in the nuclear
region of the galaxy centre. We find that the sBHs can grow up to several tens
solar mass and form heavier sBH binaries, which will be helpful for us to
understand the black-hole mass distribution as observed by the current and
future ground-based gravitational wave detectors\,(e.g., LIGO/VIRGO, ET and
Cosmic Explorer). We further estimate the event rate of extreme mass-ratio
inspirals\,(EMRI) for sBH surrounding the massive black hole and calculate the
stochastic gravitational wave\,(GW) background of the EMRIs. We find that the
background can be detected in future space-borne GW detectors after considering
the sBHs embedded in the AGN disk, while the mass accretion has a slight effect
on the GW background.
