Deep factorisation of the stable process II: potentials and applications

Here, we propose a different perspective of the deep factorisation in (Electron. J. Probab. 21 (2016) Paper No. 23, 28) based on determining potentials. Indeed, we factorise the inverse of the MAP-exponent associated to a stable process via the Lamperti-Kiu transform. Here our factorisation is completely independent from the derivation in (Electron. J. Probab. 21 (2016) Paper No. 23, 28), moreover there is no clear way to invert the factors in (Electron. J. Probab. 21 (2016) Paper No. 23, 28) to derive our results. Our method gives direct access to the potential densities of the ascending and descending ladder MAP of the Lamperti-stable MAP in closed form. In the spirit of the interplay between the classical Wiener-Hopf factorisation and the fluctuation theory of the underlying Lévy process, our analysis will produce a collection of new results for stable processes. We give an identity for the law of the point of closest reach to the origin for a stable process with index α € (0, 1) as well as an identity for the the law of the point of furthest reach before absorption at the origin for a stable process with index α€ (1, 2). Moreover, we show how the deep factorisation allows us to compute explicitly the limiting distribution of stable processes multiplicatively reflected in such a way that it remains in the strip [-1, 1].