Anomalous diffusion, infinite measures, and limit distributions in a class of exactly solvable stochastic processes

We consider stochastic processes, piecewise constant in time, which can be viewed as a gapless sequences of statistically independent box functions of duration τ_i and height v_i. Lengths and heights of the boxes are deterministically coupled, whereas the areas τ_iv_i are i.i.d. random variables. These processes are closely related to the velocity process of space-time coupled Levy walks [1,2], but can more generally be viewed as a renewal processes, where the life times τ_i are not simply drawn from one given distribution, but are determined via a deterministic law by the values v_i of a relevant stochastic variable. Despite its simplicity, these processes can show anomalous diffusion, or, more general, anomalous behavior of its moments. The latter can be related either to invariant densities, which are non-normalizable, or, for other parameters, to certain scaling densities. We obtained exact results for these quantities because we were able to derive the exact form of the propagator analytically. In addition, we were able to derive non-trivial exact limit distributions for time-averaged quantities of interest.
[1] M. Shlesinger, B. West and J. Klafter, Phys. Rev. Lett. 58, 1100 (1987)
[2] T. Albers and G. Radons, Phys. Rev. Lett. 120, 104501 (2018)