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.


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