Lévy meets Boltzmann: strange initial conditions for Brownian and fractional Fokker–Planck equations

We study normal and anomalous diffusion processes with initial conditions of the broad Lévy type, i.e., with such initial conditions which, per se, exhibit a diverging variance. In the force-free case, the behaviour of the associated probability density function features distinct shoulders which can be related to the probability current flowing away from the origin. In the presence of an external potential which eventually leads to the emergence of a non-trivial, normalisable equilibrium probability density function, the initially diverging variance becomes finite. In particular, the effects of strange initial conditions for the harmonic Ornstein–Uhlenbeck potential are explored to some detail. Methods to quantify the dynamics related to such kinds of processes are investigated.