Equivalence of Liouville measure and Gaussian free field

Abstract

Given an instance $h$ of the Gaussian free field on a planar domain $D$ and a constant $\gamma \in (0,2)$, one can use various regularization procedures to make sense of the Liouville quantum gravity area measure $\mu := e^{\gamma h(z)} dz.$ It is known that the field $h$ a.s. determines the measure $\mu_h$. We show that the converse is true: namely, $h$ is measurably determined by $\mu_h$. More generally, given a random closed fractal subset $\mathcal A$ endowed with a Frostman measure $\sigma$ whose support is $\mathcal A$ (independent of $h$), a Gaussian multiplicative chaos measure $\mu_{\sigma,h}$ can be constructed. We give a mild condition on $(\mathcal A,\sigma)$ under which $\mu_{\sigma,h}$ determines $h$ restricted to $\mathcal A$, in the sense that it determines its harmonic extension off $\mathcal A$. Our condition is satisfied by the occupation measures of planar Brownian motion and SLE curves under natural parametrizations. Along the way we obtain general positive moment bounds for Gaussian multiplicative chaos. Contrary to previous results, this does not require any assumption on the underlying measure $\sigma$ such as scale invariance, and hence may be of independent interest.