Critical Gaussian multiplicative chaos for singular measures

Given d≥1, we provide a construction of the random measure – the critical Gaussian Multiplicative Chaos – formally defined as e2dXdμ where X is a log-correlated Gaussian field and μ is a locally finite measure on Rd. Our construction generalizes the one performed in the case where μ is the Lebesgue measure. It requires that the measure μ is sufficiently spread out, namely that for μ almost every x we have ∫B(x,1)μ(dy)|x−y|deρlog1|x−y|<∞, where ρ:R+→R+ can be chosen to be any lower envelope function for the 3-Bessel process (this includes ρ(x)=xα with α∈(0,1/2)). We prove that three distinct random objects converge to a common limit which defines the critical GMC: the derivative martingale, the critical martingale, and the exponential of the mollified field. We also show that the above criterion for the measure μ is in a sense optimal.