A Concise Course on Stochastic Partial Differential Equations
Prévôt, Claudia.
A Concise Course on Stochastic Partial Differential Equations [electronic resource] / by Claudia Prévôt, Michael Röckner. - VI, 148 p. online resource. - Lecture Notes in Mathematics, 1905 0075-8434 ; . - Lecture Notes in Mathematics, 1905 .
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.
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.
9783540707813
10.1007/978-3-540-70781-3 doi
Global analysis (Mathematics).
Differential equations, partial.
Distribution (Probability theory.
Analysis.
Partial Differential Equations.
Probability Theory and Stochastic Processes.
QA299.6-433
515
A Concise Course on Stochastic Partial Differential Equations [electronic resource] / by Claudia Prévôt, Michael Röckner. - VI, 148 p. online resource. - Lecture Notes in Mathematics, 1905 0075-8434 ; . - Lecture Notes in Mathematics, 1905 .
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.
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.
9783540707813
10.1007/978-3-540-70781-3 doi
Global analysis (Mathematics).
Differential equations, partial.
Distribution (Probability theory.
Analysis.
Partial Differential Equations.
Probability Theory and Stochastic Processes.
QA299.6-433
515