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.