Multiplicative chaos and random translation
type: Item
instance of: scholarly article
title: Multiplicative chaos and random translation
zbMATH Open document ID: 0806.60038
author: Masakazu Tamashiro
author: Hiroshi Sato
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1994-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77482
review text: Let \((\Omega, {\mathcal F}, {\mathcal P})\) be a probability space and \((T, \Sigma, \sigma)\) be a triple made of a compact metric space \(T\), its Borel field and a finite measure. Let \(X_ k\) denote a sequence of zero- mean, measurable and independent Gaussian processes with respective covariances \(\Gamma_ k\). \(M_ n\) is a process defined by the equality \[ M_ n (\omega,t) = e^{\sum^ n_{k = 1} [X_ k (\omega,t) - {1 \over 2} \Gamma_ k (t,t)]}. \] For continuous functions \(\varphi\), on \(T\), one can define a random measure on \(\Sigma\), \(M_ \sigma\), by the equality \[ \int_ T \varphi (t) M_ \sigma (\omega, dt) = \lim_ n \int_ T \varphi (t) M_ n (\omega, t) \sigma (dt). \] The map \(\sigma \to M_ \sigma\) is a multiplicative chaos. \(\sigma\) is \(M\)-regular if \(E [M_ \sigma (T)] = \sigma(T)\), and \(M\)-singular if \(E [M_ \sigma (T)] = 0\). The authors of the paper show that the study of regularity and singularity of multiplicative chaos can be profitably carried out by reducing the problem to that of the equivalence and singularity of particular Gaussian sequences. They are able, in particular, to disprove a conjecture of \textit{J.-P. Kahane} concerning the regularity of multiplicative chaos [Harmonic analysis, Proc. Conf., Sendai/Jap. 1990, ICM-90 Satell. Conf. Proc., 125-139 (1991; Zbl 0768.60013)].
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60G30
Mathematics Subject Classification ID: 60G40
zbMATH DE Number: 579559
zbMATH Keywords: random measure
zbMATH Keywords: multiplicative chaos
zbMATH Keywords: equivalence and singularity of particular Gaussian sequences
reviewed by: A. F. Gualtierotti
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 579559