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Abstract
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 selfsimilar 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 nullrecurrent extension of the stable process killed on exiting a cone, showing that it again remains in the class of selfsimilar 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 selfsimilar Markov process, combined with its special status as a Lévy processes having semitractable potential analysis.
Original language  English 

Pages (fromto)  20662099 
Number of pages  34 
Journal  Annales Henri Poincare 
Volume  57 
Issue number  4 
Early online date  20 Oct 2021 
DOIs  
Publication status  Published  30 Nov 2021 
Keywords
 Duality
 Entrance law
 Kelvin transform
 Lévy processes
 Stable processes
ASJC Scopus subject areas
 Statistics and Probability
 Statistics, Probability and Uncertainty
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 2 Finished

Self Similarity and Stable Processes
Kyprianou, A.
Engineering and Physical Sciences Research Council
1/10/14 → 30/03/16
Project: Research council

RealValued SelfSimilar Markov Processes and their Applications
Kyprianou, A.
Engineering and Physical Sciences Research Council
2/06/14 → 1/10/17
Project: Research council