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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 [10,13]. In particular, we turn our attention to the case of d-dimensional isotropic stable process, for d ≥ 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. [3], 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.where the two terms either side of the multiplication sign constitute the two Wiener-Hopf factors. See e.g. Chapter VI in [2] for background. Recall that if Ψ is the characteristic exponent of any Lévy process, then there exist two Bernstein functions κ andκ (see [18] for a definition) such that, up to a multiplicative constant,Identity (1.7) is what we refer to as the Wiener-Hopf factorisation. The left-hand factor codes the range of the running maximum and the right-hand factor codes the range of the running 2 infimum of ξ. It can be checked that both belong to the class of so-called beta subordinators (see [8], as well as some of the discussion later in this paper) and, in particular, have infinite activity. This implies that ξ is regular for both the upper and lower half-lines, which in turn, means that any sphere of radius r > 0 is regular for both its interior and exterior for X. This and the fact that X has càdlàg paths ensures that, denotingthe quantity X G(t) is well defined as the point of closest reach to the origin up to time t in the sense that X G(t)− = X G(t) andThe process (G(t), t ≥ 0) is monotone increasing and hence there is no problem defining G(∞) = lim t→∞ G(t) almost surely. Moreover, as X is transient in the sense of (1.3), it is also clear that, almost surely, G(∞) = G(t) for all t sufficiently large and thatOur first main result provides explicitly the law of X G(∞) .Theorem 1.1 (Point of Closest Reach to the origin). The law of the point of closest reach to the origin is given by P x (X G(∞) ∈ dy) = π −d/2 Γ (d/2) 2 Γ ((d − α)/2) Γ (α/2) (|x| 2 − |y| 2 ) α/2 |x − y| d |y| α dy, 0 < |y| < |x|.

Bañuelos and Bogdan [6] and Bogdan et al. [19] analyse the asymptotic tail distribution of the first time a stable (Lévy) process in dimension d ≥ 2 exists a cone. We use these results to develop the notion of a stable process conditioned to remain in a cone as well as the the notion of a stable process conditioned to absorb continuously at the apex of a cone (without leaving the cone). As self-similar Markov processes we examine some of their fundamental properties through the lens of its Lamperti-Kiu decomposition. In particular we are interested to understand the underlying structure of the Markov additive process that drives such processes. As a consequence of our interrogation of the underlying MAP, we are able to provide an answer by example to the open question: If the modulator of a MAP has a stationary distribution, under what conditions does its ascending ladder MAP have a stationary distribution?With the help of an analogue of the Riesz-Bogdan-Żak transform (cf. Bogdan and Żak [20], Kyprianou [40], Alili et al. [1]) as well as Hunt-Nagasawa duality theory, we show how the two forms of conditioning are dual to one another. Moreover, in the sense of Rivero [52,53] and Fitzsimmons [32], we construct the null-recurrent extension of the stable process killed on exiting a cone, showing that it again remains in the class of self-similar Markov processes. Aside from the Riesz-Bogdan-Żak transform and Hunt-Nagasawa duality, an unusual combination of the Markov additive renewal theory of e.g. Alsmeyer [2] as well as the boundary Harnack principle (see e.g. [19]) play a central role to the analysis.In the spirit of several very recent works (see [44,40,43,45,46,41]), the results presented here show that many previously unknown results of stable processes, which have long since been understood for Brownian motion, or are easily proved for Brownian motion, become accessible by appealing to the notion of the stable process as a self-similar Markov process, in addition to its special status as a Lévy processes with a semi-tractable potential analysis.