High order asymptotic expansion for Wiener functionals

TL;DR

This paper develops a high-order asymptotic expansion theory for Wiener functionals using Malliavin calculus and Fourier methods, providing detailed characteristic and density expansions for vector-valued random variables.

Contribution

It introduces a novel high-order asymptotic expansion framework combining Malliavin calculus with Fourier techniques for Wiener functionals.

Findings

Derived explicit formulas for characteristic functionals and densities

Analyzed a wave equation example with space-time white noise

Revealed correlation structures of solutions to stochastic PDEs

Abstract

By combining the Malliavin calculus with Fourier techniques, we develop a high-order asymptotic expansion theory for a sequence of vector-valued random variables. Our asymptotic expansion formulas give the development of the characteristic functional and of the local density of the random vectors up to an arbitrary order. We analyzed in details an example related to the wave equation with space-time white noise which also provides interesting facts on the correlation structure of the solution to this equation.