On the exponential decay of oscillatory integrals on an abstract Wiener space
type: Item
instance of: scholarly article
title: On the exponential decay of oscillatory integrals on an abstract Wiener space
zbMATH Open document ID: 0914.60016
author: Setsuo Taniguchi
published in: Journal of Functional Analysis
publication date: 1999-06-20T00:00:00Z
review text: 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 \(I(\lambda)\).
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 1216767
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: oscillatory integral
zbMATH Keywords: exponential decay
zbMATH Keywords: saddle point method
zbMATH Keywords: analytic Wiener functionals
reviewed by: Jacques Franchi
MaRDI profile type: Publication
full work available at URL: jfan.1997.3185
OpenAlex ID: W2084442547
cites work: Finite dimensional approximation approach to oscillatory integrals and stationary phase in infinite dimensions
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cites work: Analytic functions, Cauchy formula, and stationary phase on a real abstract Wiener space
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cites work: Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels
DOI: 10.1006/JFAN.1997.3185
description: scientific article; zbMATH DE number 1216767