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In this revised and extended version of his course notes from a 1year 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 CameronMartin 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 semigroups, paying special attention to their longtime behavior: ergodicity, invariant measure. Here fundamental results like the theorems of Prokhorov, Von Neumann, KrylovBogoliubov and Khas'minski are proved. The last chapter is devoted to gradient systems and their asymptotic behavior.
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