EAN: 9783319341194

ADVANCED COURSES IN MATHEMATICS - CRM BARCELONA...

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. Inturn  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.

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