  In this paper, we continue our understanding of the stable process from the
perspective of the theory of self-similar Markov processes in the spirit of the
recent papers of Kyprianou (2016) and Kyprianou et al. (2017). In particular,
we turn our attention to the case of $d$-dimensional isotropic stable process,
for $d\geq 2$. Using a completely new approach we consider the distribution of
the point of closest reach. This leads us to a number of other substantial new
results for this class of stable processes. We engage with a new radial
excursion theory, never before used, from which we develop the classical
Blumenthal-Getoor-Ray identities for first entry/exit into a ball, cf.
Blumenthal et al. (1961), to the setting of $n$-tuple laws. We identify
explicitly the stationary distribution of the stable process when reflected in
its running radial supremum. Moreover, we provide a representation of the
Wiener-Hopf factorisation of the MAP that underlies the stable process through
the Lamperti-Kiu transform.
