Diffusive processes run with non-linear clocks

Brownian motion, fractional Brownian motion (fBm) and Levy motion are stochastic processes with stationary increments that have been used to study diffusion as well as a number of other natural phenomena such as conductivity fields and landscape formations. We construct a family of processes which generalize these processes through the use of nonstationary increments. This family preserves some of the underlying properties such as the fractal dimension while modifying other properties such as the mean square displacement. The explicit construction of these processes and a study of their properties demonstrate a number of misconceptions found in the literature on anomalous diffusion. In the case of Brownian motion, it demonstrates that, contrary to what has been stated in the literature, a power-law mean square displacement is not necessarily related to a breakdown in the central limit theorem. In the case of fBm, it demonstrates that the p-variation test proposed in the literature is not adequate to differentiate between difusive processes and that there may be issues with the definition of anomalous diffusion itself. The case of Levy motion further drives home the latter point. These processes are useful in their own right for modeling diffusive processes, because they are flexible enough to accommodate a wide range of behaviors. We also use these to motiviate a class of random fields which can be used to generate, e.g., terrain or conductivity fields