  Recent theoretical and numerical developments supported by observational
evidence strongly suggest that many globular clusters host a black hole (BH)
population in their centers. This stands in contrast to the prior long-standing
belief that a BH subcluster would evaporate after undergoing core collapse and
decoupling from the cluster. In this work, we propose that the inhomogeneous
Brownian motion generated by fluctuations of the stellar gravitational field
may act as a mechanism adding a stabilizing pressure to a BH population. We
argue that the diffusion equation for Brownian motion in an inhomogeneous
medium with spatially varying diffusion coefficient and temperature, which was
first discovered by Van Kampen, also applies to self-gravitating systems.
Applying the stationary phase space probability distribution to a single BH
immersed in a Plummer globular cluster, we infer that it may wander as far as
$\sim 0.05,\,0.1,\,0.5{\rm pc}$ for a mass of $m_{\rm b} \sim
10^3,\,10^2,\,10{\rm M}_\odot$, respectively. Furthermore, we find that the
fluctuations of a fixed stellar mean gravitational field are sufficient to
stabilize a BH population above the Spitzer instability threshold.
Nevertheless, we identify an instability whose onset depends on the Spitzer
parameter, $S = (M_{\rm b}/M_\star) (m_{\rm b}/m_\star)^{3/2} ,$ and parameter
$B = \rho_{\rm b}(0) (4\pi r_c^3/M_b)(m_\star/m_{\rm b})^{3/2} $, where
$\rho_{\rm b}(0)$ is the Brownian population central density. For a Plummer
sphere, the instability occurs at $(B,S) = (140,0.25)$. For $B > 140,$ we get
very cuspy BH subcluster profiles that are unstable with regard to the support
of fluctuations alone. For $S > 0.25,$ there is no evidence of any stationary
states for the BH population based on the inhomogeneous diffusion equation.

Astrophysics of Galaxies;High Energy Astrophysical Phenomena;Statistical Mechanics