High order asymptotic expansion for Wiener functionals?

Combining the Malliavin calculus with Fourier techniques, we develop a high-order asymptotic expansion theory for general Wiener functionals. Our method gives an expansion of the characteristic functional and of the local density of a Wiener functional up to an arbitrary order. The asymptotic expansion is distributional. Except for the non-degeneracy of the limit covariance matrix, we do not assume any condition of non-degeneracy of the Malliavin covariance like a non-degeneracy condition for temporally local characteristic functions so far assumed in the theory for mixing processes, that corresponds to the Cramer condition in the classical setting. Moreover, our method does not require the Markovian property used in the mixing approach. An application to the stochastic wave equation with space-time white noise is discusses. & COPY; 2023 Published by Elsevier B.V.

Automated summary

By combining the Malliavin calculus with Fourier techniques, this study develops a high-order asymptotic expansion theory for general Wiener functionals. The method allows for an expansion of the characteristic function and local density of a Wiener functional up to an arbitrary order, and the expansion is distributional. Unlike previous research, this method does not require non-degeneracy conditions or the Markovian property.