Some elements on Lévy processes. (English) Zbl 0982.60042

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].


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