Expansion of the density: A Wiener-chaos approach. (English) Zbl 0924.60030

Márquez-Carreras, David; Sanz-Solé, Marta

Bernoulli 5, No. 2, 257-274 (1999).

A Wiener functional with Wiener chaos decomposition \(F^\varepsilon=y+\sum_{n=1}^\infty\varepsilon^nI_n(f_n)\) depends on a small parameter \(\varepsilon\). Assuming that the variable \(F=F^1\) belongs to appropriate Sobolev spaces, it is proved that \(F^\varepsilon\) is smooth with respect to \(\varepsilon\). Then, under a nondegeneracy condition on the Malliavin matrix of \(F^\varepsilon\), an expansion for the density \(p^\varepsilon\) taken at the mean value \(y\) is obtained. Finally, this result is applied to two classes of hyperbolic stochastic partial differential equations with small noise.

Reviewer: J.Picard (Aubière)

MSC:

60H07	Stochastic calculus of variations and the Malliavin calculus
60H15	Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

Malliavin calculus; probability densities; stochastic partial differential equations; Wiener functionals