Lognormal star-scale invariant random measures

Allez, Romain; Rhodes, Rémi; Vargas, Vincent

Allez, Romain; Rhodes, Rémi; Vargas, Vincent (2013), Lognormal star-scale invariant random measures, Probability Theory and Related Fields, 155, 3-4, p. 751-788. http://dx.doi.org/10.1007/s00440-012-0412-9

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Type

Article accepté pour publication ou publié

Date

2013

Journal name

Probability Theory and Related Fields

Volume

155

Number

3-4

Publisher

Springer

Pages

751-788

Abstract (EN)

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multplicative chaos theory developped by J.P. Kahane in 1985 (or possibly extensions of this theory). As a by product, we also obtain an explicit characterization of the covariance structure of these measures. We also prove that qualitative properties such as long-range independence or isotropy can be read off the equation.

Subjects / Keywords

Scale invariance; Random measure; Multiplicative chaos; Star equation; Uniqueness