The inverse stable subordinator provides a probability model for time-fractional differential
equations, and leads to explicit solution formulae. This paper reviews properties of the …

Fractional calculus is a rapidly growing field of research, at the interface between probability,
differential equations, and mathematical physics. It is used to model anomalous diffusion, in …

Lévy processes are the natural continuous-time analogue of random walks and form a rich
class of stochastic processes around which a robust mathematical theory exists. Their …

Lévy processes are the natural continuous-time analogue of random walks and form a rich
class of stochastic processes around which a robust mathematical theory exists. Their …

This paper aims at unifying and clarifying the recent advances in the analysis of the
fractional and generalized fractional Partial Differential Equations of Caputo and Riemann …

A continuous-time random walk is a simple random walk subordinated to a renewal process
used in physics to model anomalous diffusion. In this paper we show that, when the time …

I first heard about long-range dependence while working on a book on stable processes
with Murad Taqqu. Initially, the notion did not seem to stand out among other notions I was …

The doubly stochastic Poisson process is a generalization of the ordinary Poisson process
in the sense that stochastic variation in the intensity is allowed. Some authors call these …

We discuss a renewal process in which successive events are separated by scale-free
waiting time periods. Among other ubiquitous long-time properties, this process exhibits …

The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its
distributions solve a time-fractional analogue of the Kolmogorov forward equation for a …