Open AccessFree transport for convex potentials
Author(s) -
Yoann Dabrowski,
Alice Guionnet,
Dimitri Shlyakhtenko
Publication year - 2021
Publication title -
new zealand journal of mathematics
Language(s) - English
Resource type - Journals
eISSN - 1179-4984
pISSN - 1171-6096
DOI - 10.53733/102
Subject(s) - convexity , construct (python library) , extension (predicate logic) , mathematics , regular polygon , tensor product , pure mathematics , commutative property , quadratic equation , state (computer science) , convex analysis , computer science , geometry , convex optimization , algorithm , financial economics , economics , programming language
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.