×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: Link
References:
[1] M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964) · Zbl 0171.38503
[2] L. Alili, L. Chaumont, P. Graczyk and T. Żak. Inversion, duality and Doob htransforms for self-similar Markov processes. Electron. J. Probab. 22, Paper No. 20, 18 (2017) · Zbl 1357.60079
[3] G. Alsmeyer. On the Markov renewal theorem. Stochastic Process. Appl. 50 (1), 37–56 (1994) · Zbl 0789.60066
[4] G. Alsmeyer. Quasistochastic matrices and Markov renewal theory. J. Appl. Probab. 51A (Celebrating 50 Years of The Applied Probability Trust), 359–376 (2014) · Zbl 1325.60140
[5] S. Asmussen. Applied probability and queues, volume 51 of Applications of Mathematics (New York). Springer-Verlag, New York, second edition (2003). ISBN 0-387-00211-1 · Zbl 1029.60001
[6] S. Asmussen and H. Albrecher. Ruin probabilities, volume 14 of Advanced Series on Statistical Science & Applied Probability. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, second edition (2010). ISBN 978-981-4282-52-9; 981-428252-9
[7] J. Bertoin. Lévy processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1996). ISBN 0-521-56243-0
[8] J. Bertoin and M.-E. Caballero.Entrance from 0+ for increasing semi-stable Markov processes. Bernoulli 8 (2), 195–205 (2002) · Zbl 1002.60032
[9] J. Bertoin and W. Werner. Stable windings. Ann. Probab. 24 (3), 1269–1279 (1996) · Zbl 0867.60045
[10] J. Bertoin and M. Yor. The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17 (4), 389–400 (2002) · Zbl 1004.60046
[11] J. Bliedtner and W. Hansen. Potential theory. Universitext. Springer-Verlag, Berlin (1986). ISBN 3-540-16396-4 · Zbl 0706.31001
[12] L. E. Blumenson. Classroom Notes: A Derivation of n-Dimensional Spherical Coordinates. Amer. Math. Monthly 67 (1), 63–66 (1960)
[13] R. M. Blumenthal and R. K. Getoor. Markov processes and potential theory. Pure and Applied Mathematics, Vol. 29. Academic Press, New York-London (1968) · Zbl 0169.49204
[14] R. M. Blumenthal, R. K. Getoor and D. B. Ray. On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99, 540–554 (1961) · Zbl 0118.13005
[15] K. Bogdan, K. Burdzy and Z.-Q. Chen. Censored stable processes. Probab. Theory Related Fields 127 (1), 89–152 (2003) · Zbl 1032.60047
[16] K. Bogdan and T. Żak. On Kelvin transformation. J. Theoret. Probab. 19 (1), 89–120 (2006) · Zbl 1105.60057
[17] J. Bretagnolle. Résultats de Kesten sur les processus à accroissements indépendants pages 21–36. Lecture Notes in Math., Vol. 191 (1971)
[18] T. Byczkowski, J. Mał ecki and M. Ryznar. Bessel potentials, hitting distributions and Green functions. Trans. Amer. Math. Soc. 361 (9), 4871–4900 (2009) · Zbl 1181.60121
[19] M. E. Caballero and L. Chaumont. Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43 (4), 967–983 (2006a) · Zbl 1133.60316
[20] M. E. Caballero and L. Chaumont. Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Probab. 34 (3), 1012–1034 (2006b) · Zbl 1098.60038
[21] M. E. Caballero, J. C. Pardo and J. L. Pérez. Explicit identities for Lévy processes associated to symmetric stable processes. Bernoulli 17 (1), 34–59 (2011) · Zbl 1284.60092
[22] E. Çinlar. Markov additive processes. I, II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24, 85–93; ibid. 24 (1972), 95–121 (1972)
[23] E. Çinlar. Lévy systems of Markov additive processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31, 175–185 (1974/75)
[24] E. Çinlar. Entrance-exit distributions for Markov additive processes. Math. Programming Stud. (5), 22–38 (1976)
[25] L. Chaumont, A. Kyprianou, J. C. Pardo and V. Rivero. Fluctuation theory and exit systems for positive self-similar Markov processes. Ann. Probab. 40 (1), 245–279 (2012) · Zbl 1241.60019
[26] L. Chaumont, A. E. Kyprianou and J. C. Pardo. Some explicit identities associated with positive self-similar Markov processes. Stochastic Process. Appl. 119 (3), 980–1000 (2009) · Zbl 1170.60017
[27] L. Chaumont, H. Pantí and V. Rivero. The Lamperti representation of real-valued self-similar Markov processes. Bernoulli 19 (5B), 2494–2523 (2013) · Zbl 1284.60077
[28] L. Chaumont and J. C. Pardo. The lower envelope of positive self-similar Markov processes. Electron. J. Probab. 11, no. 49, 1321–1341 (2006) · Zbl 1127.60034
[29] L. Chaumont and V. Rivero.On some transformations between positive selfsimilar Markov processes. Stochastic Process. Appl. 117 (12), 1889–1909 (2007) · Zbl 1129.60039
[30] S. Dereich, L. Döring and A. E. Kyprianou. Real self-similar processes started from the origin. Ann. Probab. 45 (3), 1952–2003 (2017) · Zbl 1372.60052
[31] R. A. Doney. Fluctuation theory for Lévy processes, volume 1897 of Lecture Notes in Mathematics. Springer, Berlin (2007). ISBN 978-3-540-48510-0; 3-540-48510-4
[32] R. A. Doney and A. E. Kyprianou. Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16 (1), 91–106 (2006) · Zbl 1101.60029
[33] L. Döring and A. E. Kyprianou. Entrance and exit at infinity for stable jump diffusions. ArXiv Mathematics e-prints (2018)
[34] E. B. Dynkin. Some limit theorems for sums of independent random variables with infinite mathematical expectations. In Select. Transl. Math. Statist. and Probability, Vol. 1, pages 171–189. Inst. Math. Statist. and Amer. Math. Soc., Providence, R.I. (1961) · Zbl 0112.10105
[35] R. K. Getoor. Continuous additive functionals of a Markov process with applications to processes with independent increments. J. Math. Anal. Appl. 13, 132–153 (1966) · Zbl 0138.40901
[36] S. E. Graversen and J. Vuolle-Apiala. α-self-similar Markov processes. Probab. Theory Relat. Fields 71 (1), 149–158 (1986) · Zbl 0561.60085
[37] B. Haas and R. Stephenson. On the exponential functional of Markov additive processes, and applications to multi-type self-similar fragmentation processes and trees. ArXiv Mathematics e-prints (2017)
[38] J. Horowitz. Semilinear Markov processes, subordinators and renewal theory. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24, 167–193 (1972) · Zbl 0251.60052
[39] J. Jacod and A. N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition (2003). ISBN 3-540-43932-3 · Zbl 1018.60002
[40] J. Janssen and R. Manca. Semi-Markov risk models for finance, insurance and reliability. Springer, New York (2007). ISBN 978-0-387-70729-7; 0-387-70729-8 · Zbl 1144.91027
[41] H. Kaspi. On the symmetric Wiener-Hopf factorization for Markov additive processes. Z. Wahrsch. Verw. Gebiete 59 (2), 179–196 (1982) · Zbl 0468.60072
[42] H. Kesten. A convolution equation and hitting probabilities of single points for processes with stationary independent increments. Bull. Amer. Math. Soc. 75, 573–578 (1969a) · Zbl 0201.19002
[43] H. Kesten. Hitting probabilities of single points for processes with stationary independent increments. Memoirs of the American Mathematical Society, No. 93. American Mathematical Society, Providence, R.I. (1969b) · Zbl 0201.19002
[44] H. Kesten. Hitting probabilities of single points for processes with stationary independent increments. Memoirs of the American Mathematical Society, No. 93. American Mathematical Society, Providence, R.I. (1969c) · Zbl 0201.19002
[45] H. Kesten. Renewal theory for functionals of a Markov chain with general state space. Ann. Probability 2, 355–386 (1974) · Zbl 0303.60090
[46] J. F. C. Kingman. Recurrence properties of processes with stationary independent increments. J. Austral. Math. Soc. 4, 223–228 (1964) · Zbl 0127.34905
[47] S. W. Kiu. Semistable Markov processes in Rn. Stochastic Process. Appl. 10 (2), 183–191 (1980) · Zbl 0436.60054
[48] A. Kuznetsov, A. E. Kyprianou, J. C. Pardo and A. R. Watson. The hitting time of zero for a stable process. Electron. J. Probab. 19, no. 30, 26 (2014) · Zbl 1293.60055
[49] A. Kuznetsov and J. C. Pardo. Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Acta Appl. Math. 123, 113–139 (2013) · Zbl 1268.60060
[50] A. E. Kyprianou. Fluctuations of Lévy processes with applications. Universitext. Springer, Heidelberg, second edition (2014). ISBN 978-3-642-37631-3; 978-3-64237632-0
[51] A. E. Kyprianou. Deep factorisation of the stable process. Electron. J. Probab. 21, Paper No. 23, 28 (2016) · Zbl 1338.60130
[52] A. E. Kyprianou and J. C. Pardo. Stable lévy processes via lamperti-type representations (2018+). Book in progress, to be published in Cambridge University Press
[53] A. E. Kyprianou, J. C. Pardo and A. R. Watson. Hitting distributions of α-stable processes via path censoring and self-similarity. Ann. Probab. 42 (1), 398–430 (2014) · Zbl 1306.60051
[54] A. E. Kyprianou and P. Patie. A Ciesielski-Taylor type identity for positive selfsimilar Markov processes. Ann. Inst. Henri Poincaré Probab. Stat. 47 (3), 917– 928 (2011) · Zbl 1231.60031
[55] A. E. Kyprianou, V. Rivero and B. Şengül. Deep factorisation of the stable process II: Potentials and applications. Ann. Inst. Henri Poincaré Probab. Stat. 54 (1), 343–362 (2018) · Zbl 1396.60040
[56] A. E. Kyprianou, V. Rivero and W. Satitkanitkul. Conditioned real self-similar Markov processes. ArXiv Mathematics e-prints (2015) · Zbl 1442.60047
[57] A. E. Kyprianou, V. Rivero and W. Satitkanitkul. Deep factorisation of the stable process III: radial excursion theory and the point of closest reach. ArXiv Mathematics e-prints (2016)
[58] A. E. Kyprianou and A. R. Watson. Potentials of stable processes. In Séminaire de Probabilités XLVI, volume 2123 of Lecture Notes in Math., pages 333–343. Springer, Cham (2014) · Zbl 1390.60174
[59] J. Lamperti. An invariance principle in renewal theory. Ann. Math. Statist. 33, 685–696 (1962) · Zbl 0106.33902
[60] J. Lamperti. Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22, 205–225 (1972) · Zbl 0274.60052
[61] N. S. Landkof.Foundations of modern potential theory.Springer-Verlag, New York-Heidelberg (1972) · Zbl 0253.31001
[62] N. N. Lebedev. Special functions and their applications. Dover Publications, Inc., New York (1972) · Zbl 0271.33001
[63] T. Luks. Boundary behavior of α-harmonic functions on the complement of the sphere and hyperplane. Potential Anal. 39 (1), 29–67 (2013) · Zbl 1277.60135
[64] E. Nane, Y. Xiao and A. Zeleke. A strong law of large numbers with applications to self-similar stable processes. Acta Sci. Math. (Szeged) 76 (3-4), 697–711 (2010) · Zbl 1274.60098
[65] P. Patie. Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes.Ann. Inst. Henri Poincaré Probab. Stat. 45 (3), 667–684 (2009) · Zbl 1180.31010
[66] P. Patie. Law of the absorption time of some positive self-similar Markov processes. Ann. Probab. 40 (2), 765–787 (2012) · Zbl 1241.60020
[67] S. C. Port. The first hitting distribution of a sphere for symmetric stable processes. Trans. Amer. Math. Soc. 135, 115–125 (1969) · Zbl 0209.49303
[68] S. C. Port and C. J. Stone. Infinitely divisible processes and their potential theory. Ann. Inst. Fourier (Grenoble) 21 (2), 157–275; ibid. 21 (1971), no. 4, 179–265 (1971) · Zbl 0195.47601
[69] S. C. Port and C. J. Stone. Brownian motion and classical potential theory. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1978). ISBN 0-12-561850-6 · Zbl 0413.60067
[70] N. U. Prabhu. Queues and inventories. A study of their basic stochastic processes. John Wiley & Sons, Inc., New York-London-Sydney (1965) · Zbl 0131.16904
[71] C. Profeta and T. Simon. On the harmonic measure of stable processes. In Séminaire de Probabilités XLVIII, volume 2168 of Lecture Notes in Math., pages 325– 345. Springer, Cham (2016) · Zbl 1367.60057
[72] P. E. Protter. Stochastic integration and differential equations, volume 21 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin (2005). ISBN 3-540-00313-4
[73] D. Ray. Stable processes with an absorbing barrier. Trans. Amer. Math. Soc. 89, 16–24 (1958) · Zbl 0083.13804
[74] M. Riesz. Intégrales de Riemann-Liouville et potentiels. Acta. Sci. Math. Szeged. 9, 1–42 (1938) · Zbl 0018.40704
[75] B. A. Rogozin. Distribution of the position of absorption for stable and asymptotically stable random walks on an interval. Teor. Verojatnost. i Primenen. 17, 342–349 (1972)
[76] K. Sato. Lévy processes and infinitely divisible distributions, volume 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2013). ISBN 978-1-107-65649-9
[77] R. L. Schilling, R. Song and Z. Vondraček. Bernstein functions, volume 37 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, second edition (2012). ISBN 978-3-11-025229-3; 978-3-11-026933-8
[78] R. Stephenson. On the exponential functional of Markov additive processes, and applications to multi-type self-similar fragmentation processes and trees. ArXiv Mathematics e-prints (2017) · Zbl 1414.60022
[79] J. Vuolle-Apiala and S. E. Graversen. Duality theory for self-similar processes. Ann. Inst. H. Poincaré Probab. Statist. 22 (3), 323–332 (1986) · Zbl 0608.60057
[80] H. Widom. Stable processes and integral equations. Trans. Amer. Math. Soc. 98, 430–449 (1961) · Zbl 0097.12905
[81] Y. Xiao.Asymptotic results for self-similar Markov processes.In Asymptotic methods in probability and statistics (Ottawa, ON, 1997), pages 323–340. NorthHolland, Amsterdam (1998) · Zbl 0936.60060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.