Negative moments for Gaussian multiplicative chaos on fractal sets

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 -infinity. 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.