From Levy-Type Processes to Parabolic SPDEs

This volume contains 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, held at the Centre de Recerca Matemŕtica (CRM) in Bellaterra (Barcelona).The course by Davar Khoshnevisan deals with some problems in the field of stochastic partial differential equations of parabolic type. More precisely, the main objective is to establish an Invariance Principle for those equations in a rather general setting, and also deduce, as an application, comparison-type results. The framework in which these problems are addressed go beyond the classical setting, in the sense that the driving noise is assumed to be a multiplicative space-time white noise on a group, and the underlying elliptic operator corresponds to a generator of a Lévy process on that group. This implies that stochastic integration with respect to the above noise, as well as existence and uniqueness of solution for the corresponding equation, become relevant on their own. These aspects are also developed, in parallel with a lot of illustrative examples.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.