Stable Lévy processes in a cone

Bañuelos and Bogdan (Potential Anal. 21 (3) (2004) 263–288) and Bogdan et al. (Electron. J. Probab. 23 (2018) 11) analyse the asymptotic tail distribution of the first time a stable (Lévy) process in dimension d ≥ 2 exits 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 (MAP) that drives such processes. Through the 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 (J. Theoret. Probab. 19 (1) (2006) 89–120), Kyprianou (Electron. J. Probab. 21 (2016) 23), Alili et al. (Electron. J. Probab. 22 (2017) 20)) 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 (Bernoulli 11 (3) (2005) 471–509; Bernoulli 13 (4) (2007) 1053–1070) and Fitzsimmons (Electron. Commun. Probab. 11 (2006) 230–241), 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 (Stochastic Process. Appl. 50 (1) (1994) 37–56) as well as the boundary Harnack principle (see e.g. Electron. J. Probab. 23 (2018) 11) play a central role to the analysis. In the spirit of several very recent works (see Stochastic Process. Appl. 129 (3) (2019) 954–977; Electron. J. Probab. 21 (2016) 23; Ann. Inst. Henri Poincaré Probab. Stat. 54 (1) (2018) 343–362; Potential Anal. 53 (2020) 1347–1375; ALEA Lat. Am. J. Probab. Math. Stat. 15 (1) (2018) 617–690; Ann. Probab. 48 (3) (2020) 1220–1265), the results presented here show that 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 properties of the stable process as a self-similar Markov process, combined with its special status as a Lévy processes having semi-tractable potential analysis.