Multiplicative chaos and random translation
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)].
equivalence and singularity of particular Gaussian sequences
scientific article; zbMATH DE number 579559