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Applications quantitatives et geometriques du calcul de Malliavin. (Quantitative and geometric applications of the Malliavin calculus). (French) Zbl 0666.60014
Stochastic analysis, Proc. Jap.-Fr. Semin., Paris/France 1987, Lect. Notes Math. 1322, 109-133 (1988).
[For the entire collection see Zbl 0635.00012.]
This is a survey article primarily on some probabilistic methods in geometry. A central idea is the heuristic notion of submersion in the weak and the strong sense. While the strong sense submersion is a geometric description, the weak sense submersion is an integrability condition. These two notions are often equivalent. The author describes some applications of the weak sense submersion. Initially he applies the technique to the asymptotics of the transition probability density. Then he applies the technique to the computation of some index theorems.
After writing this paper the author discovered the work of this reviewer who was the first to use probability in the computation of local formulae for some index theorems. He notes this work at the end of the references. Finally the author applies the weak sense submersion tothe asymptotics of the density for a diffusion whose generator is hypoelliptic.
Reviewer: T.E.Duncan

60D05 Geometric probability and stochastic geometry
60H07 Stochastic calculus of variations and the Malliavin calculus
60J60 Diffusion processes
58J65 Diffusion processes and stochastic analysis on manifolds