This book gives the basis of the probabilistic functional analysis on Wiener space, developed during the last decade. The subject has progressed considerably in recent years thr- ough its links with QFT and the impact of Stochastic Calcu- lus of Variations of P. Malliavin. Although the latter deals essentially with the regularity of the laws of random varia- bles defined on the Wiener space, the book focuses on quite different subjects, i.e. independence, Ramer's theorem, etc. First year graduate level in functional analysis and theory of stochastic processes is required (stochastic integration with respect to Brownian motion, Ito formula etc). It can be taught as a 1-semester course as it is, or in 2 semesters adding preliminaries from the theory of stochastic processes It is a user-friendly introduction to Malliavin calculus!
Preliminaries -- Gross-Sobolev derivative, divergence and Ornstein-Uhlenbeck operator -- Meyer inequalities -- Hypercontractivity -- L p -multipliers theorem, meyer inequalities and distributions -- Some applications of the distributions -- Positive distributions and applications -- Characterization of independence of some Wiener functionals -- Moment inequalities for Wiener functional -- to the theorem of Ramer
Probability Theory and Stochastic Processes
Theoretical Mathematical and Computational Physics