  Around the 1960s a celebrated collection of papers emerged offering a number
of explicit identities for the class of isotropic stable processes in one and
higher dimensions; these include, for example, the lauded works of Blumenthal,
Getoor, Ray, Port and Rogozin. Amongst other things, these results nicely
exemplify the use of standard Riesz potential theory on the unit open ball
$\mathbb{B}_d :=\{ x\in \mathbb{R}^d :|x|<1\}$,
$\mathbb{R}^d\backslash\mathbb{B}_d$ and $\mathbb{S}_d :=\partial \mathbb{B}_d$
with the, then, modern theory of potential analysis for Markov processes.
Following initial observations of Lamperti in 1972, with the occasional
sporadic work of Kiu, Vuolle-Apiala and Graversen in the 1980s, an alternative
understanding of stable processes through the theory of self-similar Markov
processes has prevailed in the last decade or more. This point of view offers
deeper probabilistic insights into some of the aforementioned potential
analytical relations.
  In this review article, we will rediscover many of the aforementioned
classical identities in relation to the unit ball and recent extensions thereof
by combining elements of these two theories, which have otherwise been largely
separated by decades in the literature. We present a dialogue that appeals as
much as possible to path decompositions. Most notable in this respect is the
Lamperti-Kiu decomposition of self-similar Markov processes and the
Riesz-Bogdan-\.Zak transformation. Some of the proofs we give are known, some
are known proofs mixed with new methods and some proofs are completely new. We
assume that the reader has a degree of familiarity with the bare basics of
L\'evy processes but nonetheless, we often include reminders of standard
material.
