  Using the DIANOGA hydrodynamical zoom-in simulation set of galaxy clusters,
we analyze the dynamics traced by stars belonging to the Brightest Cluster
Galaxies (BCGs) and their surrounding diffuse component, forming the
intracluster light (ICL), and compare it to the dynamics traced by dark matter
and galaxies identified in the simulations. We compute scaling relations
between the BCG and cluster velocity dispersions and their corresponding masses
(i.e. $M_\mathrm{BCG}^{\star}$- $\sigma_\mathrm{BCG}^{\star}$, $M_{200}$-
$\sigma_{200}$, $M_\mathrm{BCG}^{\star}$- $M_{200}$,
$\sigma_\mathrm{BCG}^{\star}$- $\sigma_{200}$), we find in general a good
agreement with observational results. Our simulations also predict
$\sigma_\mathrm{BCG}^{\star}$- $\sigma_{200}$ relation to not change
significantly up to redshift $z=1$, in line with a relatively slow accretion of
the BCG stellar mass at late times. We analyze the main features of the
velocity dispersion profiles, as traced by stars, dark matter, and galaxies. As
a result, we discuss that observed stellar velocity dispersion profiles in the
inner cluster regions are in excellent agreement with simulations. We also
report that the slopes of the BCG velocity dispersion profile from simulations
agree with what is measured in observations, confirming the existence of a
robust correlation between the stellar velocity dispersion slope and the
cluster velocity dispersion (thus, cluster mass) when the former is computed
within $0.1 R_{500}$. Our results demonstrate that simulations can correctly
describe the dynamics of BCGs and their surrounding stellar envelope, as
determined by the past star-formation and assembly histories of the most
massive galaxies of the Universe.
