On the exponential decay of oscillatory integrals on an abstract Wiener space
The author presents here a continuation of the big article, by himself and \textit{P. Malliavin} [J. Funct. Anal. 143, No. 2, 470-528 (1997; Zbl 0873.60040)]. His aim in the present work is to get a better estimate for a stochastic oscillatory integral \(I(\lambda)=\int e^{\sqrt {-1}\lambda q(x)} \psi (x) \mu(dx)\) as \(\lambda\to\infty\), where \(q\) and \(\psi\) are real analytic Wiener functionals which satisfy some integrability conditions with respect to the Wiener measure \(\mu\). The method, as in the joint work with P. Malliavin, where however \(\psi\) was taken constant, is to mimic the saddle point method which is classical in finite dimensions. So it consists in deforming the contour integration to change the Fourier type integral \(I(\lambda)\) into an explicit Laplace integral, from which the estimate directly follows, showing an exponential decay for \(I(\lambda)\).
scientific article; zbMATH DE number 1216767