We consider log-correlated random fields $X$ and the associated multiplicative chaos measures $\mu_{ \gamma,X}$. Our results reconstruct the underlying field $X$ from the multiplicative chaos measure $\nu_{ \gamma,X}$. The new feature of our results is that we allow the dimension $d$ to be arbitrary and cover also the critical case $\gamma=\sqrt{ 2d}$. In the sub-critical regime $\gamma<\sqrt{ 2d}$, we allow the fields to be mildly non-Gaussian, that is, the field has the decomposition $X=G+H$ with a log-correlated Gaussian field $G$ and a H\"older-continuos (not necessarily Gaussian) field $H$.
Categories: math.PR math-ph math.MP