Electronic Communications in Probability

Negative moments for Gaussian multiplicative chaos on fractal sets

Abstract

The objective of this note is to study the probability that the total mass of a subcritical Gaussian multiplicative chaos (GMC) with arbitrary base measure $\sigma$ is small. When $\sigma$ has some continuous density w.r.t Lebesgue measure, a scaling argument shows that the logarithm of the total GMC mass is sub-Gaussian near $-\infty$. However, when $\sigma$ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure $\sigma$, the total GMC mass has negative moments of all orders.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 100, 10 pp.

Dates
Accepted: 11 September 2018
First available in Project Euclid: 19 December 2018

https://projecteuclid.org/euclid.ecp/1545188960

Digital Object Identifier
doi:10.1214/18-ECP168

Mathematical Reviews number (MathSciNet)
MR3896838

Zentralblatt MATH identifier
07023489

Citation

Garban, Christophe; Holden, Nina; Sepúlveda, Avelio; Sun, Xin. Negative moments for Gaussian multiplicative chaos on fractal sets. Electron. Commun. Probab. 23 (2018), paper no. 100, 10 pp. doi:10.1214/18-ECP168. https://projecteuclid.org/euclid.ecp/1545188960

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