Abstract
Bañuelos and Bogdan (2004) and Bogdan, Palmowski and Wang (2016) 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 selfsimilar Markov processes we examine some of their fundamental properties through the lens of its LampertiKiu 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?
We show how the two forms of conditioning are dual to one another. Moreover, we construct the nullrecurrent extension of the stable process killed on exiting a cone, showing that it again remains in the class of selfsimilar Markov processes.
In the spirit of several very recent works, 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 selfsimilar Markov process, in addition to its special status as a Lévy processes with a semitractable potential analysis.
We show how the two forms of conditioning are dual to one another. Moreover, we construct the nullrecurrent extension of the stable process killed on exiting a cone, showing that it again remains in the class of selfsimilar Markov processes.
In the spirit of several very recent works, 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 selfsimilar Markov process, in addition to its special status as a Lévy processes with a semitractable potential analysis.
Original language  English 

Journal  Annales Henri Poincare 
Publication status  Acceptance date  13 Dec 2020 
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Profiles

Andreas Kyprianou
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Probability Laboratory at Bath
 Institute for Mathematical Innovation (IMI)  Director of the Bath Institute for Mathematical Innovation
Person: Research & Teaching, Teaching & Other