zbMATH — the first resource for mathematics

Propriétés d’absolue continuité dans les espaces de Dirichlet et application aux équations différentielles stochastiques. (Absolut continuity properties in Dirichlet spaces and applications to stochastic differential equations). (French) Zbl 0642.60044
Probabilités XX, Proc. Sémin., Strasbourg 1984/85, Lect. Notes Math. 1204, 131-161 (1986).
[For the entire collection see Zbl 0593.00014.]
A Lipschitzian functional calculus is valid in Dirichlet spaces for the Dirichlet form and its “carré du champ” operator; this is proved for the univariate calculus by the first author, C. R. Acad. Sci., Paris, Sér. I 298, 133-136 (1984; Zbl 0594.31018), in the locally compact case, and by the authors, J. Funct. Anal. 69, 229-259 (1986; Zbl 0605.60058), on general measurable spaces. Here we extend these results to a multivariate calculus in the case of the Dirichlet space of the Ornstein-Uhlenbeck semigroup on the Wiener space.
This is done by establishing a general density criterion for multivariate random ariables: the “density property of the image of the energetic volume”. The application to Lipschitzian S.D.E. goes through a crucial factorization lemma and leads us to three theorems on existence of densities for the laws of the solutions of S.D.E., including the uniformly degenerated case.

60H99 Stochastic analysis
60J60 Diffusion processes
60B99 Probability theory on algebraic and topological structures
Full Text: Numdam EuDML