On Wiener functionals of order 2 associated with soliton solutions of the KdV equation

The eigenvalues and eigenvectors of the Hilbert–Schmidt operators corresponding to the Wiener functionals of order 2, which give a rise of soliton solutions of the KdV equation, are determined. Two explicit expressions of the stochastic oscillatory integral with such Wiener functional as phase function are given; one is of infinite product type and the other is of Lévyʹs formula type. As an application, the asymptotic behavior of the stochastic oscillatory integral will be discussed.