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Stable Lévy processes, self-similarity and the unit ball. (English) Zbl 1396.60051
Summary: Around the 1960s a celebrated collection of papers emerged offering a number of explicit identities for the class of isotropic Lévy stable processes in one and higher dimensions; these include, for example, the lauded works of D. Ray [Trans. Am. Math. Soc. 89, 16–24 (1958; Zbl 0083.13804)]; H. Widom [Trans. Am. Math. Soc. 98, 430–449 (1961; Zbl 0097.12905)]; B. A. Rogozin [Theory Probab. Appl. 17, 332–338 (1972; Zbl 0272.60050); translation from Teor. Veroyatn. Primen. 17, 342–349 (1972)] (in one dimension) and R. M. Blumenthal et al. [Trans. Am. Math. Soc. 99, 540–554 (1961; Zbl 0118.13005)]; R. K. Getoor [J. Math. Anal. Appl. 13, 132–153 (1966; Zbl 0138.40901)]; S. C. Port [Trans. Am. Math. Soc. 135, 115–125 (1969; Zbl 0209.49303)] (in higher dimension), see also more recently T. Byczkowski et al. [Trans. Am. Math. Soc. 361, No. 9, 4871–4900 (2009; Zbl 1181.60121)]; T. Luks [Potential Anal. 39, No. 1, 29–67 (2013; Zbl 1277.60135)]. Amongst other things, these results nicely exemplify the use of standard Riesz potential theory on the unit open ball $$\mathbb B_d:=\{x\in\mathbb R^d:|x|<1\}$$, $$\mathbb R^d\backslash \mathbb B_d$$ and the sphere $$\mathbb S_{d-1}:=\{x\in\mathbb R^d:|x|=1\}$$ with the, then, modern theory of potential analysis for Markov processes.
Following initial observations of J. Lamperti [Z. Wahrscheinlichkeitstheor. Verw. Geb. 22, 205–225 (1972; Zbl 0274.60052)], with the occasional sporadic contributions such as S.-W. Kiu [Stochastic Processes Appl. 10, 183–191 (1980; Zbl 0436.60054)]; J. Vuolle-Apiala and S. E. Graversen [Ann. Inst. Henri Poincaré, Probab. Stat. 22, 323–332 (1986; Zbl 0608.60057)]; S. E. Graversen and J. Vuolle-Apiala [Probab. Theory Relat. Fields 71, 149–158 (1986; Zbl 0561.60085)], an alternative understanding of Lévy stable processes through the theory of self-similar Markov processes has prevailed in the last decade or more. This point of view offers deeper probabilistic insights into some of the aforementioned potential analytical relations; see for example J. Bertoin and M. Yor [Potential Anal. 17, No. 4, 389–400 (2002; Zbl 1004.60046)]; J. Bertoin and M.-E. Caballero [Bernoulli 8, No. 2, 195–205 (2002; Zbl 1002.60032)]; M. E. Caballero and L. Chaumont [J. Appl. Probab. 43, No. 4, 967–983 (2006; Zbl 1133.60316); Ann. Probab. 34, No. 3, 1012–1034 (2006; Zbl 1098.60038)]; L. Chaumont et al. [Stochastic Processes Appl. 119, No. 3, 980–1000 (2009; Zbl 1170.60017)]; P. Patie [Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 3, 667–684 (2009; Zbl 1180.31010); Ann. Probab. 40, No. 2, 765–787 (2012; Zbl 1241.60020]; K. Bogdan and T. Żak [J. Theor. Probab. 19, No. 1, 89–120 (2006; Zbl 1105.60057)]; A. E. Kyprianou et al. [Ann. Probab. 42, No. 1, 398–430 (2014; Zbl 1306.60051)]; A. Kuznetsov et al. [Electron. J. Probab. 19, Paper No. 30, 26 p. (2014; Zbl 1293.60055)]; A. E. Kyprianou and A. R. Watson [Lect. Notes Math. 2123, 333–343 (2014; Zbl 1390.60174)]; A. Kuznetsov and J. C. Pardo [Acta Appl. Math. 123, No. 1, 113–139 (2013; Zbl 1268.60060)]; A. E. Kyprianou [Electron. J. Probab. 21, Paper No. 23, 28 p. (2016; Zbl 1338.60130)]; A. E. Kyprianou et al. [Ann. Inst. Henri Poincaré, Probab. Stat. 54, No. 1, 343–362 (2018; Zbl 1396.60040); “Deep factorisation of the stable process III: radial excursion theory and the point of closest reach”, Preprint, arXiv:1706.09924]; L. Alili et al. [Electron. J. Probab. 22, Paper No. 20, 18 p. (2017; Zbl 1357.60079)].
In this review article, we will rediscover many of the aforementioned classical identities in relation to the unit ball by combining elements of these two theories, which have otherwise been largely separated by decades in the literature. We present a dialogue that appeals as much as possible to path decompositions. Most notable in this respect is the Lamperti-Kiu decomposition of self-similar Markov processes given in [Kiu loc. cit.; Chaumont et al. loc. cit.; Alili et al. loc. cit.] and the Riesz-Bogdan-Żak transformation given in [Bogdan and Żak; loc. cit.].
Some of the results and proofs we give are known (marked $$\heartsuit$$), some are mixed with new results or methods, respectively, (marked $$\diamondsuit$$) and some are completely new (marked $$\clubsuit$$). We assume that the reader has a degree of familiarity with the bare basics of Lévy processes but, nonetheless, we often include reminders of standard material.

MSC:
 60G51 Processes with independent increments; Lévy processes 60G18 Self-similar stochastic processes 60G52 Stable stochastic processes
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