Lognormal scale invariant random measures

Romain Allez (1) , Rémi Rhodes (1) , Vincent Vargas (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multiplicative chaos theory developed 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.

Mots clés: Random measure; Star equation; Scale invariance; Multiplicative chaos; Uniqueness

Domaines: Probabilités [math.PR]

Romain Allez, Rémi Rhodes, Vincent Vargas. Lognormal scale invariant random measures. Probability Theory and Related Fields, 2012, 155 (3-4), pp.751-788. ⟨hal-00561713v3⟩