  Concepts like `typicality' and the `eigenstate thermalization hypothesis' aim
at explaining the apparent equilibration of quantum systems, possibly after a
very long time. However, these concepts are not concerned with the specific way
in which this equilibrium is approached. Our point of departure is the
(evident) observation that some forms of the approach to equilibrium, such as,
e.g., exponential decay of observables, are much more common then others. We
suggest to trace this dominance of certain decay dynamics back to a larger
stability with respect to generic Hamiltonian perturbations. A numerical study
of a number of examples in which both, the unperturbed Hamiltonians as well as
the perturbations are modeled by partially random matrices is presented. We
furthermore develop a simple heuristic, mathematical scheme that describes the
result of the numerical investigations remarkably well. According to those
investigations the exponential decay indeed appears to be most stable. Dynamics
that are in a certain sense at odds with the arrow of time are found to be very
unstable.
