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

Abstract

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 $0<\alpha \le 2$. 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{é}vy flights in shallow external potentials.

22 pages, 12 figures

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