Anomalous diffusion schemes underlying the Cole–Cole relaxation: The role of the inverse-time α-stable subordinator

The paper presents the random-variable formalism of the anomalous diffusion processes. The emphasis is on a rigorous presentation of asymptotic behaviour of random walk processes with infinite mean random time intervals between jumps. We elucidate the role of the so-called inverse-time stochastic process, the main mathematical tool that allows us to modify the dynamics of standard relaxation processes and give rise to the nonexponential decay of modes. In particular, we show that the Brownian motion in combination with an appropriate inverse-time process may lead not only to exponential but also to the nonexponential relaxation responses.