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Lognormal star-scale invariant random measures

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
Publication identifier
http://dx.doi.org/10.1007/s00440-012-0412-9
Metadata
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Author(s)
Allez, Romain
Rhodes, Rémi
Vargas, Vincent
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

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