An Introduction to Infinite-Dimensional Analysis

• Κύριος συγγραφέας: Prato, Giuseppe Da (Συγγραφέας)
• Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
• Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Berlin, Heidelberg : Springer Berlin Heidelberg, 2006.
• Θέματα: Mathematics.; Functional analysis.; Partial differential equations.; Probabilities.; Functional Analysis.; Probability Theory and Stochastic Processes.; Partial Differential Equations.

 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. 

 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.