The main theme of this Monograph is the study of degenerate stochastic differential equations, considered as transformations of the Wiener measure, and their relationship with partial differential equations. The book contains an elementary derivation of Malliavin's integration by parts formula, a proof of the probabilistic form of Hormander's theorem, an extension of Hormander's theorem for infinitely degenerate differential operators, and criteria for the regularity of measures induced by stochastic hereditary-delay equations.

Preface List of notations Introduction 1. Background material 2. Regularity of measures induced by stochastic differential equations 3. The probabilistic form of Hormander's theorem 4. Infinitely degenerate hypoelliptic operators 5. Smooth densities for a class of stochastic functional equations 6. Generalized divergence operators and absolutely continuous transformations References

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