Subordinators: Examples and applications. (English) Zbl 0955.60046

Bertoin, Jean

Bertoin, J. (ed.) et al., Lectures on probability theory and statistics. Ecole d’eté de Probabilités de Saint-Flour XXVII-1997, Saint-Flour, France, July 7-23, 1997. Berlin: Springer. Lect. Notes Math. 1717, 1-91 (1999).

The aim of this paper is to present a survey of methods, results and applications in the theory of subordinators (or increasing Lévy processes valued on the half-line), which reflects one of the author’s preferred subjects. It can be seen as a very useful (and also somewhat easier) complement to his now classical book “Lévy processes” (1996; Zbl 0861.60003), even though it can also be read independently of the latter. The leitfaden of the article is the one-to-one correspondence between subordinators and regenerative (or Markov) sets. This makes it possible to apply the sample path and statistical properties of the former (which is the matter of the 1st part of the article) to a better geometric understanding of the latter (which concerns the 2nd part). Typically, a Markov set is the set of times where some Markov process visits some fixed time of its state space, so that the relatively simple theory of subordinators allows some fine study of processes which are much more general and complicated. In spite of its shortness, the material covered in this article is very rich, so that it seems reasonable to the reviewer to give only a sketch of its content.
In the first two chapters the author presents the basic theory (Lévy-Khinchin formula, range of a subordinator, regenerative sets, connection with Markov processes and their local times at regular points). In the next three chapters he develops a finer study, which concerns in particular the asymptotic behaviour of last-passage times [including the celebrated Dynkin-Lamperti theorem, see J. Lamperti, Ann. Math. Stat. 33, 685-696 (1962; Zbl 0106.33902) and E. B. Dynkin, Select. Transl. Math. Stat. Probab. 1, 171-189 (1961); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 19, No. 4, 247-266 (1955; Zbl 0068.12402)], the smoothness of local times [including Fristedt-Pruitt’s law of the iterated logarithm see B. E. Fristedt and W. E. Pruitt, Z. Wahrscheinlichkeitstheorie Verw. Geb. 18, 167-182 (1971; Zbl 0197.44204)], and the geometry of Markov sets [with the results of J. Hawkes on their Hausdorff dimension and independent intersection, see ibid. 33, 113-132 (1975; Zbl 0402.60074) and ibid. 37, 243-251 (1977; Zbl 0404.60077)]. The last four chapters are devoted to the applications of the above theory to various problems in stochastic analysis, and include numerous contributions of the author himself. These chapters are mainly concerned with a statistical (Brownian) study of the inviscid Burgers equation [see the author, Commun. Math. Phys. 193, No. 2, 397-406 (1998; Zbl 0917.60063)], a thickness problem in random covering [see P. J. Fitzsimmons, B. Fristedt and L. A. Shepp, Z. Wahrscheinlichkeitstheorie Verw. Geb. 70, 175-189 (1985; Zbl 0564.60008)], local times at a fixed point and at the supremum for Lévy processes [including the work of R. A. Doney and the author of Spitzer’s condition, Ann. Inst. Henri Poincaré, Probab. Stat. 33, No. 2, 167-178 (1997; Zbl 0880.60078)], and, last, the occupation time process of a linear diffusion (of Brownian motion in particular).
For the entire collection see [Zbl 0930.00052].

Reviewer: Thomas Simon (Berlin)

MSC:

60G51	Processes with independent increments; Lévy processes
60D05	Geometric probability and stochastic geometry
60H30	Applications of stochastic analysis (to PDEs, etc.)
60J25	Continuous-time Markov processes on general state spaces
60J55	Local time and additive functionals
60J60	Diffusion processes
60J65	Brownian motion
60K05	Renewal theory

Keywords:

subordinator; regenerative set; renewal theory; last passage time; law of the iterated logarithm; fractal dimensions; Burgers equation; random covering; ladder time process; occupation time process; Lévy process; Markov process