Expansion of the density: A Wiener-chaos approach (Q1290375)

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.