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An Introduction to Infinite-Dimensional Analysis [electronic resource] / by Giuseppe Da Prato.

By: Prato, Giuseppe Da [author.].
Contributor(s): SpringerLink (Online service).
Material type: TextTextSeries: Universitext: Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2006Description: X, 208 p. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9783540290216.Subject(s): Functional analysis | Distribution (Probability theory | Differential equations, partial | Functional Analysis | Probability Theory and Stochastic Processes | Partial Differential EquationsAdditional physical formats: Printed edition:: No title; Printed edition:: No title; Printed edition:: No titleDDC classification: 515.7 Online resources: Click here to access online
Contents:
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: Springer eBooksSummary: 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.
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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.

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Introduction to infinite-dimensional analysis by Da Prato Giuseppe
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