Now in its second edition, this book gives a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. In the first part the authors give a self-contained exposition of the basic properties of probability measure on separable Banach and Hilbert spaces, as required later; they assume a reasonable background in probability theory and finite dimensional stochastic processes. The second part is devoted to the existence and uniqueness of solutions of a general stochastic evolution equation, and the third concerns the qualitative properties of those solutions. Appendices gather together background results from analysis that are otherwise hard to find under one roof. This revised edition includes two brand new chapters surveying recent developments in the area and an even more comprehensive bibliography, making this book an essential and up-to-date resource for all those working in stochastic differential equations.
- Thoroughly updated to reflect changes since publication of the first edition
- Provides a solid foundation to the whole theory of stochastic evolution equations
- Useful starting point for further research
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Reviews & endorsements
Review of the first edition: 'The exposition is excellent and readable throughout, and should help bring the theory to a wider audience.' Daniel L. Ocone, Stochastics and Stochastic Reports
Review of the first edition: '… a welcome contribution to the rather new area of infinite dimensional stochastic evolution equations, which is far from being complete, so it should provide both a useful background and motivation for further research.' Yuri Kifer, The Annals of Probability
Review of the first edition: '… an excellent book which covers a large part of stochastic evolution equations with clear proofs and a very interesting analysis of their properties … In my opinion this book will become an indispensable tool for everybody working on stochastic evolution equations and related areas.' P. Kotelenez, American Mathematical Society
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Product details
- Edition: 2nd Edition
- Date Published: April 2014
- format: Hardback
- isbn: 9781107055841
- length: 512 pages
- dimensions: 236 x 152 x 33 mm
- weight: 0.9kg
- availability: Available
Table of Contents
Preface
Introduction
Part I. Foundations:
1. Random variables
2. Probability measures
3. Stochastic processes
4. Stochastic integral
Part II. Existence and Uniqueness:
5. Linear equations with additive noise
6. Linear equations with multiplicative noise
7. Existence and uniqueness for nonlinear equations
8. Martingale solutions
9. Markov property and Kolmogorov equation
10. Absolute continuity and Girsanov theorem
11. Large time behavior of solutions
12. Small noise asymptotic
13. Survey of specific equations
14. Some recent developments
Appendix A. Linear deterministic equations
Appendix B. Some results on control theory
Appendix C. Nuclear and Hilbert–Schmidt operators
Appendix D. Dissipative mappings
Bibliography
Index.
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Authors
Giuseppe Da Prato, Scuola Normale Superiore, Pisa
Giuseppe Da Prato is Emeritus Professor at the Scuola Normale Superiore di Pisa. His research activity concerns: stochastic analysis, evolution equations both deterministic and stochastic, elliptic and parabolic equations with infinitely many variables, deterministic and stochastic control. On these subjects he has produced more than 350 papers in reviewed journals and eight books.
Jerzy Zabczyk, Polish Academy of Sciences
Jerzy Zabczyk is Professor in the Institute of Mathematics at the Polish Academy of Sciences. His research interests include stochastic processes, evolution equations, control theory and mathematical finance. He has published 87 papers in mathematical journals and seven books.