Title: Free transport for convex potentials,
Abstract: We construct non-commutative analogs of transport maps among free Gibbs state
satisfying a certain convexity condition. Unlike previous constructions, our
approach is non-perturbative in nature and thus can be used to construct
transport maps between free Gibbs states associated to potentials which are far
from quadratic, i.e., states which are far from the semicircle law. An
essential technical ingredient in our approach is the extension of free
stochastic analysis to non-commutative spaces of functions based on the
Haagerup tensor product.