Simplified Malliavin calculus. (English) Zbl 0609.60066

Norris, James

Probabilités XX, Proc. Sémin., Strasbourg 1984/85, Lect. Notes Math. 1204, 101-130 (1986).

[For the entire collection see Zbl 0593.00014.]
A careful exposition is given of a probabilistic proof of (part of) Hörmander’s theorem on the hypoellipticity of second order partial differential operators. This uses the Malliavin calculus in the formulation of Bismut. There are several simplifications of earlier work, notably in the proof of a semimartingale inequality of Kusuoka and Stroock which underlies the relation of the non-degeneracy of the Malliavin covariance matrix to Lie brackets formed from the coefficients of the associated stochastic differential equation.

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60	Diffusion processes
60G30	Continuity and singularity of induced measures
60G46	Martingales and classical analysis

Keywords:

Hörmander’s theorem on the hypoellipticity of second order partial differential operators; Malliavin calculus; semimartingale inequality; Lie brackets

Citations:

Zbl 0593.00014