An introduction to Malliavin’s calculus. (English) Zbl 0546.60055

Stochastic analysis, Proc. Taniguchi Int. Symp., Katata & Kyoto/Jap. 1982, North-Holland Math. Libr. 32, 1-52 (1984).
[For the entire collection see Zbl 0538.00017.]
This is a very detailed and exhaustive introduction to the stochastic calculus of variation for Wiener functionals with a lot of hints to related papers and applications.
The authors started with the finite dimensional situation. They introduce in this case the Ornstein-Uhlenbeck operator and the corresponding machinery and obtain first criteria for the smoothness of image measures. They then use the Gross’ scheme of an abstract Wiener space and consider Wiener functionals and Sobolev spaces. In this frame many theorems on the existence of a density of image measures and smoothness properties are obtained. In a 15 page long subsection the technique is applied to Wiener functionals stemming from stochastic differential equations.
There is also a short discussion on several related topics as stochastic oscillatory integrals, capacities, filtering, infinite dimensional diffusions, boundary value problems and de Rham complexes of Wiener functionals. The list of references contains 96 items.
Reviewer: M.Breger


60Hxx Stochastic analysis
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes


Zbl 0538.00017