Around the 1960s a celebrated collection of papers emerged offering a number of explicit
identities for the class of isotropic Lévy stable processes in one and higher dimensions; …

Stable Lévy processes lie at the intersection of Lévy processes and self-similar Markov
processes. Processes in the latter class enjoy a Lamperti-type representation as the space …

We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random
search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation …

Since the seminal work of Lamperti, there is a lot of interest in the understanding of the
general structure of self-similar Markov processes. Lamperti gave a representation of …

We show that any R^d∖{0\}-valued self-similar Markov process X, with index α>0 can be
represented as a path transformation of some Markov additive process (MAP) (θ,ξ) in S_d …

Abstract The Lamperti–Kiu transformation for real-valued self-similar Markov processes
(rssMp) states that, associated to each rssMp via a space-time transformation, there is a …

A Markov Additive Process is a bi-variate Markov process $(\xi, J)=\big ((\xi_t, J_t),
t\geq0\big) $ which should be thought of as a multi-type L\'evy process: the second …

In his seminal work from the 1950s, William Feller classified all onedimensional diffusions
on−∞≤ a< b≤∞ in terms of their ability to access the boundary (Feller's test for explosions) …

In recent work, Chaumont et al.(2013) showed that is possible to condition a stable process
with index α∈(1, 2) to avoid the origin. Specifically, they describe a new Markov process …

This note surveys some recent results on self-similar Markov processes. Since the research
around the topic has been very rich during the last fifteen years we do not pretend to cover …