Applications of the Malliavin calculus. III. (English) Zbl 0633.60078

The paper is a continuation of the authors earlier works on the Malliavin calculus [Stochastic analysis, Proc. Taniguchi Int. Symp., Katata & Kyoto/Jap. 1982, North Holland Math. Libr. 32, 271-306 (1984; Zbl 0546.60056) and J. Fac. Sci. Univ. Tokyo, Sect. I A 32, 1-76 (1985; Zbl 0568.60059)]. Using impressive technical machinery the authors establish the smoothness of transition functions associated with the solution of a Stratonovich stochastic integral equation. Under certain conditions they show the transition function has a density which is bounded above and below by ‘Gaussian kernels’ in which the Euclidean metric is replaced by a ‘control metric’ defined in terms of the coefficient vector fields.
If L is the second order operator associated with the solution process the authors also give a quantitative Harnack principle and a Poincaré inequality for L.
Reviewer: R.J.Elliot


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