This volume presents the lecture notes from two courses given by Davar
Khoshnevisan and René Schilling, respectively, at the second Barcelona
Summer School on Stochastic Analysis.
René Schilling's notes are an expanded version of his course on Lévy and
Lévy-type processes, the purpose of which is two-fold: on the one hand,
the course presents in detail selected properties of the Lévy processes,
mainly as Markov processes, and their different constructions,
eventually leading to the celebrated Lévy-Itô decomposition. On the
other, it identifies the infinitesimal generator of the Lévy process as
a pseudo-differential operator whose symbol is the characteristic
exponent of the process, making it possible to study the properties of
Feller processes as space inhomogeneous processes that locally behave
like Lévy processes. The presentation is self-contained, and includes
dedicated chapters that review Markov processes, operator semigroups,
random measures, etc.
In turn, Davar Khoshnevisan's course investigates selected 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 to
deduce, as an application, comparison-type results. The framework in
which these problems are addressed goes 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
the existence and uniqueness of a solution for the corresponding
equation, become relevant in their own right. These aspects are also
developed and supplemented by a wealth of illustrative examples.
