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

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.

### MSC:

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