On dimensional reduction of functional measures
type: Item
instance of: scholarly article
title: On dimensional reduction of functional measures
zbMATH Open document ID: 0644.28011
DOI: 10.1016/0022-1236(88)90094-8
author: Asao Arai
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: For functional measures indexed by rectangles in \({\mathbb{R}}^{\nu}\), dimensional reduction means (loosely speaking) that, when one side of the rectangle goes to zero, the conveniently scaled functional integrals converge to the functional integrals corresponding to the limit rectangle in \({\mathbb{R}}^{\nu -1}\). A simple example is the Gaussian measure of mean \(\lambda\in {\mathbb{R}}\) and covariance \((-\Delta +1)^{-1},\) where \(\Delta\) is the Laplacian on the given rectangle with periodic boundary conditions, whose characteristic functional on functions independent of one variable is simply related to the characteristic functional of the corresponding measure in lower dimension. A sufficient condition for dimensional reducibility is given for a class of functional measures absolutely continuous with respect to a Gaussian measure. The condition is shown to hold in \(P(\Phi)_{\nu}\) and ``exponential type'' models of Euclidean field theory, wherefrom limit theorems for the partition functions of the models in two-dimensional space-time follow.
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 46G12
Mathematics Subject Classification ID: 81T08
zbMATH DE Number: 4049935
zbMATH Keywords: exponential type models of Euclidean field theory
zbMATH Keywords: functional measures
zbMATH Keywords: dimensional reduction
zbMATH Keywords: functional integrals
reviewed by: Nicolae Angelescu
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90094-8
OpenAlex ID: W2018579946
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description: scientific article; zbMATH DE number 4049935