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＜E-Book＞
An Introduction to Infinite-Dimensional Analysis / by Giuseppe Da Prato
(Universitext. ISSN:21916675)

Edition	1st ed. 2006.
Publisher	(Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer)
Year	2006
Language	English
Size	X, 208 p : online resource
Authors	*Da Prato, Giuseppe author SpringerLink (Online service)
Subjects	LCSH:Functional analysis LCSH:Probabilities LCSH:Differential equations FREE:Functional Analysis FREE:Probability Theory FREE:Differential Equations
Notes	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 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 HTTP:URL=https://doi.org/10.1007/3-540-29021-4

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