Random fields as generalized white noise functionals
type: Item
instance of: scholarly article
title: Random fields as generalized white noise functionals
zbMATH Open document ID: 0812.60051
DOI: 10.1007/BF00994911
author: Takeyuki Hida
published in: Acta Applicandae Mathematicae
publication date: 1995-05-14T00:00:00Z
review text: Consider a generalized white noise functional, i.e., a vector field \(\{X(C),\;C \in \mathbb{C}\}\) with \(\mathbb{C} = \{C \mid C =\) simple, closed, convex manifold diffeomorphic to the sphere \(S^{d-1}\}\). It is assumed that \(X(C) \in S^* \supset L^ 2 \supset S\), where \(L^ 2 =\) Hilbert space of ordinary white noise functionals while \(S,S^*\) denote a test functional space and its dual, respectively. Let \(\mu_ C\) be a white noise measure on \(E(C)^* \supset L^ 2 (C,d \sigma) \supset E(C) =\) nuclear space induced by a Gaussian measure \(\mu\) on \(R^ d\) with the characteristic functional \(\exp (-{1 \over 2} \| \xi \|^ 2)\). The author's main objective is to describe a stochastic variational equation for generalized white noise functionals of the form \[ X(C) = \Phi \Bigl( \int_ CF \bigl( C,x(u),u \bigr) du \Bigr),\quad x \in E^* (\mu). \] For a class of quantum-mechanical, Schrödinger type, operators \(A\) (= polynomials of degree \(\leq 2\) in \(\partial_ t\), \(\partial^*_ t\), \(t \in R^ d)\) \(X(C)\) was found to be a solution to the variational equation \({\delta X (C) \over \delta C} (\rho) = AX(C)\), \(\rho \in C\), which is seen as an extension of an analogous classical result from the theory of diffusion processes \(\{X(t),\;t \geq 0\}\).
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 22E65
zbMATH DE Number: 627451
zbMATH Keywords: convex manifold diffeomorphic to the sphere
zbMATH Keywords: nuclear space induced by a Gaussian measure
zbMATH Keywords: stochastic variational equation
zbMATH Keywords: generalized white noise functionals
zbMATH Keywords: diffusion processes
reviewed by: A. Korzeniowski
MaRDI profile type: Publication
cites work: The Radiation Theories of Tomonaga, Schwinger, and Feynman
cites work: A note on generalized Gaussian random fields
cites work: Q4202615
cites work: Conformal invariance of white noise
cites work: Infinite dimensional rotations and Laplacians in terms of white noise calculus
cites work: Brownian functionals and applications
cites work: White noise approach to Gaussian random fields
cites work: Q4764551
cites work: Discrete approximation to operators in white noise analysis
cites work: On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields*
description: scientific article; zbMATH DE number 627451