A Concise Course on Stochastic Partial Differential Equations

Claudia Prévôt, Michael Röckner (auth.)3-540-70780-8, 978-3-540-70780-6

These lectures concentrate on (nonlinear) stochastic partial differential equations (SPDE) of evolutionary type. All kinds of dynamics with stochastic influence in nature or man-made complex systems can be modelled by such equations.
To keep the technicalities minimal we confine ourselves to the case where the noise term is given by a stochastic integral w.r.t. a cylindrical Wiener process.But all results can be easily generalized to SPDE with more general noises such as, for instance, stochastic integral w.r.t. a continuous local martingale.

There are basically three approaches to analyze SPDE: the “martingale measure approach”, the “mild solution approach” and the “variational approach”. The purpose of these notes is to give a concise and as self-contained as possible an introduction to the “variational approach”. A large part of necessary background material, such as definitions and results from the theory of Hilbert spaces, are included in appendices.

Table of contents :
Front Matter….Pages V-VI
Motivation, Aims and Examples….Pages 1-4
Stochastic Integral in Hilbert Spaces….Pages 5-42
Stochastic Differential Equations in Finite Dimensions….Pages 43-54
A Class of Stochastic Differential Equations….Pages 55-103
Back Matter….Pages 105-148