Deep factorisation of the stable process II: Potentials and applications

作者：Kyprianou Andreas E^*; Rivero Victor; Sengul Bati

来源：Annales de l Institut Henri Poincare-Probabilites et Statistiques, 2018, 54(1): 343-362.

DOI：10.1214/16-AIHP806

• Stable processes
• Self-similar Markov processes
• Wiener-Hopf factorisation
• Radial reflection

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 Levy 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 alpha is an element of (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 alpha is an element of (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].

• 出版日期2018-2