Some elements on Lévy processes. (English) Zbl 0982.60042

Bertoin, Jean

Shanbhag, D. N. (ed.) et al., Stochastic processes: Theory and methods. Amsterdam: North-Holland/ Elsevier. Handb. Stat. 19, 117-144 (2001).

This text is a brief survey about Lévy processes. It gives a dozen and a half of important theorems of the theory, most of which proofs can be found in the recent textbooks by the author [“Lévy processes” (1996; Zbl 0861.60003)] and K. I. Sato [“Lévy processes and infinitely divisible distributions” (1999; Zbl 0973.60001)]. These theorems are concerned with the following topics (in order of appearance): integral functionals, renewal theory (with Dynkin-Lamperti theorem), fluctuation theory (with Wiener-Hopf factorization), recurrence and transience (with Spitzer-Rogozin’s criterion, Erickson’s and Chung-Fuchs’ tests), sample path regularity (with Perkins’ theorem on slow points for stable processes), and geometric properties of the range (with Evans-Fitzsimmons-Salisbury’s theorem on multiple points).
For the entire collection see [Zbl 0961.60001].

Reviewer: Thomas Simon (Berlin)

MSC:

60G51	Processes with independent increments; Lévy processes
60D05	Geometric probability and stochastic geometry
60G17	Sample path properties
60G52	Stable stochastic processes
60J25	Continuous-time Markov processes on general state spaces

Citations:

Zbl 0861.60003; Zbl 0938.60005; Zbl 0973.60001