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

Absence of confinement and non-Boltzmann stationary states of fractional Brownian motion in shallow external potentials