Exact expressions of the stochastic oscillatory integrals with the phase function, which is the quadratic Wiener functional obtained from the Malliavin derivative of the square norm of the Brownian sample path on interval, are given. As an application, the density function of the distribution of the half of the Wiener functional is given.
Setsuo Taniguchi. "On the Quadratic Wiener Functional Associated with the Malliavin Derivative of the Square Norm of Brownian Sample Path on Interval." Electron. Commun. Probab. 11 1 - 10, 2006. https://doi.org/10.1214/ECP.v11-1174