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Calcul stochastique et opérateurs dégénérés du second ordre. I: Résolvantes, théorème de Hörmander et applications. (Stochastic calculus and degenerated second order operators. I: Resolvents, theorem of Hörmander and applications). (French) Zbl 0715.60064

The paper is the first part of three ones devoted to the development of Malliavin calculus and applications. The first section consists of the review on stochastic flows generated by stochastic differential equations and the review on Malliavin calculus for SDEs. Results on smoothness of transition density in backward and forward variables and time-dependent estimates are given. The main part of the paper is devoted to investigations of Markov resolvents smoothness, the important problem that must be done. The last sections contain applications to classical Hörmander hypoellipticity theorems for elliptic and parabolic operators of second order, the “hypoellipticity decomposition” theorem, the invariant measures smoothness for SDEs on a compact variety and conditions of the reversibility of the corresponding Markov process.
Reviewer: A.Yu.Veretennikov

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60J35 Transition functions, generators and resolvents
65H10 Numerical computation of solutions to systems of equations
35J25 Boundary value problems for second-order elliptic equations
35G15 Boundary value problems for linear higher-order PDEs
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