An Introduction to Infinite-Dimensional Analysis
Springer Berlin (Verlag)
978-3-642-42168-6 (ISBN)
In this revised and extended version of his course notes from a 1-year course at Scuola Normale Superiore, Pisa, the author provides an introduction - for an audience knowing basic functional analysis and measure theory but not necessarily probability theory - to analysis in a separable Hilbert space of infinite dimension.
Starting from the definition of Gaussian measures in Hilbert spaces, concepts such as the Cameron-Martin formula, Brownian motion and Wiener integral are introduced in a simple way. These concepts are then used to illustrate some basic stochastic dynamical systems (including dissipative nonlinearities) and Markov semi-groups, paying special attention to their long-time behavior: ergodicity, invariant measure. Here fundamental results like the theorems of Prokhorov, Von Neumann, Krylov-Bogoliubov and Khas'minski are proved. The last chapter is devoted to gradient systems and their asymptotic behavior.
GIUSEPPE DA PRATO was born in La Spezia in 1936. Having graduated in Physics in 1960 from the University of Rome, he became full professor of Mathematics in 1968 and taught in Rome and in Trento. Since 1979 he has been Professor of Mathematical Analysis at the Scuola Normale Superiore di Pisa.
Gaussian measures in Hilbert spaces.- The Cameron-Martin formula.- Brownian motion.- Stochastic perturbations of a dynamical system.- Invariant measures for Markov semigroups.- Weak convergence of measures.- Existence and uniqueness of invariant measures.- Examples of Markov semigroups.- L2 spaces with respect to a Gaussian measure.- Sobolev spaces for a Gaussian measure.- Gradient systems.
From the reviews:
"This is an extended version of the author's 'An introduction to infinite-dimensional analysis' published by Scuola Normale Superiore, Pisa ... . A well written textbook (even an introductory research monograph), suitable for teaching a graduate course." (Neils Jacob, Zentralblatt MATH, Vol. 1109 (11), 2007)
"The present volume collects together ... the notes of the course on infinite-dimensional analysis held by the author at the Scuola Normale Superiore of Pisa in recent years. The book is intended for people who have some knowledge of functional analysis ... . It provides an extremely useful tool for those scholars who are interested in learning some basics about Gaussian measures in Hilbert spaces, Brownian motion, Markov transition semigroups ... . The book is well written and all arguments are clearly and rigorously presented." (Sandra Cerrai, Mathematical Reviews, Issue 2009 a)
| Erscheint lt. Verlag | 30.11.2014 |
|---|---|
| Reihe/Serie | Universitext |
| Zusatzinfo | X, 208 p. |
| Verlagsort | Berlin |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 349 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Schlagworte | Brownian motion • Functional Analysis • Gaussian measures • hilbert space • invariant measures • Markov Processes • measure theory • Partial differential equations • Probability Theory • Sobolev Space • White Noise |
| ISBN-10 | 3-642-42168-7 / 3642421687 |
| ISBN-13 | 978-3-642-42168-6 / 9783642421686 |
| Zustand | Neuware |
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