Expansion of the density: A Wiener-chaos approach
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.
stochastic partial differential equations
scientific article; zbMATH DE number 1294460