An Introduction to Infinite-Dimensional Analysis
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Beschreibung
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.
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. Moreover, some details have been added as well as some new material on dynamical systems with dissipative nonlinearities and asymptotic behavior for gradient systems.
Based on well-known lectures given at SNS Pisa The author is a world leader in the field of infinite dimensional analysis, and the teacher of many other leaders. Has published very little in book for thus far Contains new material on dynamical systems with dissipative nonlinearities, and asymptotic behavior for gradient systems Includes supplementary material: sn.pub/extras
Autor*in
Giuseppe Da Prato
Themen in »An Introduction to Infinite-Dimensional Analysis«
Brownian motion Gaussian measures Hilbert space Markov processes Probability theory Sobolev space White noise functional analysis invariant measures measure theory partial differential equations
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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)
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