The present note is based on a series of lectures given by the author at the 35th summer school in probability at Saint-Flour in July 2005. This note is devoted to some interesting recent developments within the theory of Lévy processes. It consists of 10 sections.
The first four chapters contain a short introduction into the general theory. For example, the Lévy-Itô decomposition is presented, subordinators are introduced, their special properties are treated and, furthermore, it deals with local times and excursions, with the ladder process as well as with the Wiener-Hopf factorization of Lévy processes.
Section 5 presents recent further developments of the latter factorization, while section 6 is devoted to creeping and related questions. Section 7 deals with Spitzer’s condition for Lévy processes and random walks. The solution of a basic question about that condition (due to J. Bertoin and the author) is given. Section 8 treats Lévy processes conditioned to stay positive, while sections 9 and 10 deal with spectrally negative Lévy processes and the small-time behavior, respectively.
The presentation in this note is very clear; every section starts with a short introduction into its topic. In most cases the author motivates the properties of Lévy processes by the corresponding ones for random walks. To read this note a basic knowledge about Lévy processes is necessary.