  We study the diffusive motion of a particle in a subharmonic potential of the
form $U(x)=|x|^c$ ($0<c<2$) driven by long-range correlated, stationary
fractional Gaussian noise $\xi_{\alpha}(t)$ with $0<\alpha\le2$. In the absence
of the potential the particle exhibits free fractional Brownian motion with
anomalous diffusion exponent $\alpha$. While for an harmonic external potential
the dynamics converges to a Gaussian stationary state, from extensive numerical
analysis we here demonstrate that stationary states for shallower than harmonic
potentials exist only as long as the relation $c>2(1-1/\alpha)$ holds. We
analyse the motion in terms of the mean squared displacement and (when it
exists) the stationary probability density function (PDF). Moreover we discuss
analogies of non-stationarity of L{\'e}vy flights in shallow external
potentials.
