## Degenerate Stochastic Differential Equations and HypoellipticityThe 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. |

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### Contents

Background material | 4 |

Regularity of measures induced by stochastic | 26 |

The probabilistic form of Hörmanders theorem | 47 |

### Other editions - View all

Degenerate Stochastic Differential Equations and Hypoellipticity Denis Bell No preview available - 1995 |

### Common terms and phrases

absolutely continuous abstract adapted analysis Applying argument Assume Banach space Borel bounded calculus chapter Choose complete conclusion condition Consider converges covariance defined definition denote density depends derivatives deterministic diffusion distribution establish estimate exists exponentially extension fact Finally finite-dimensional fixed follows formula function Furthermore Gaussian given gives Hence holds Hörmander's hypoelliptic hypotheses implies independent inductive inequality integral interval invertible Iterating Itô Lebesgue measure Lemma lower bounds Malliavin matrix measure Note obtain operator parabolic particular path positive constants probability proof of Theorem prove random variable regularity relation Remark respect result satisfies sequence shown similar smooth solution standard stochastic differential equation Stroock sufficiently Suppose term test function Theorem 4.2 transformation unique University vector fields Wiener process Wiener space yields