  We review the issue of steady spherically symmetric accretion onto a
renormalization group improved Schwarzschild space-time which is solution to an
asymptotically safe theory (AS) containing high-derivative terms. We use a
Hamiltonian dynamical system approach for the analysis of the accretion of four
types of isothermal test fluids: ultra-stiff fluid, ultra-relativistic fluid,
radiation fluid, and sub-relativistic fluid. An important outcome of our study
is that, contrary to what claimed in a recent work, there exist physical
solutions for the accretion of an ultra-relativistic fluid in AS which include
subsonic, supersonic and transonic regimes. We also study quantum corrections
to the known stability of the accretion in general relativity (GR). To this end
we use a perturbative procedure based on the continuity equation with the mass
accretion rate being the perturbed quantity. Two classes of perturbations are
studied: standing waves and travelling waves. We find that quantum gravity
effects either enhance or diminish the stability of the accretion depending on
the type of test fluid and depending on the radial distance to the central
object.
