From Lévy-Type Processes to Parabolic SPDEs

This volume presents the lecture notes of the two courses given by Davar Khoshnevisan and René Schilling, respectively, at the second edition of the Barcelona Summer School on Stochastic Analysis.

René Schilling's notes are an expanded version of his course on Lévy and Lévy-type processes. Its purpose is two-fold: on the one hand, it extensively presents some properties of the Lévy processes, mainly as Markov processes, and its different constructions, leading eventually to the celebrated Lévy-Itô decomposition. On the other hand, it identifies the infinitesimal generator of the Lévy process as a pseudo-differential operator whose symbol is the characteristic exponent of the process, allowing to study the properties of Feller processes as space inhomogeneous processes that behave locally like Lévy processes. The presentation is self-contained, and different chapters are enclosed to review Markov processes, operator semigroups, random measures, etc.