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An Introduction to Infinite-Dimensional Analysis

  • Giuseppe Da Prato

About this book

Introduction

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.

Keywords

Brownian motion Gaussian measures Hilbert space Markov processes Probability theory Sobolev space White noise functional analysis invariant measures measure theory

Authors and affiliations

  • Giuseppe Da Prato
    • 1
  1. 1.Scuola Normale SuperiorePisaItaly

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