A Concise Course on Stochastic Partial Differential Equations (Lecture Notes in Mathematics) 〈Vol. 1905〉

Prévôt, Claudia/ Röckner, Michael

Springer（2007/06発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 149 p.
• 言語 ENG
• 商品コード 9783540707806
• DDC分類 515

Full Description

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.

Contents

Motivation, Aims and Examples.- Stochastic Integral in Hilbert spaces.- Stochastic Differential Equations in Finite Dimensions.- A Class of Stochastic Differential Equations in Banach Spaces.- Appendices: The Bochner Integral.- Nuclear and Hilbert-Schmidt Operators.- Pseudo Invers of Linear Operators.- Some Tools from Real Martingale Theory.- Weak and Strong Solutions: the Yamada-Watanabe Theorem.- Strong, Mild and Weak Solutions.