A Concise Course on Stochastic Partial Differential Equations

✍ Scribed by Claudia Prévôt, Michael Röckner (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2007
• Tongue: English
• Leaves: 149
• Series: Lecture notes in mathematics 1905
• Edition: 1
• Category: Library

✦ Synopsis

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

✦ Subjects

Partial Differential Equations; Probability Theory and Stochastic Processes