Free Moment Measures and Laws

In arXiv:1304.0630, it was shown that convex, almost everywhere continuous functions coordinatize a broad class of probability measures on $\mathbb{R}^n$ by the map $U \mapsto (\nabla U)_{\#} e^{-U} dx$. We consider whether there is a similar coordinatization of non-commutative probability spaces, with the Gibbs measure $e^{-U} dx$ replaced by the corresponding free Gibbs law. We call laws parametrized in this way free moment laws. We first consider the case of a single (and thus commutative) random variable and then the regime of $n$ non-commutative random variables which are perturbations of freely independent semi-circular variables. We prove that free moment laws exist with little restriction for the one dimensional case, and for small even perturbations of free semi-circle laws in the general case.