The existence of Lipschitz transport maps between probability measures leads to transportation of functional inequalities from the source to the target measure. Finding such maps, however, is a non-trivial problem. We construct a new Lipschitz transport map, the Brownian transport map, which transports the infinite dimensional Wiener measure onto measures on Euclidean spaces. I will talk about the construction of the Brownian transport map, its new Lipschitz properties, and the applications to Euclidean functional inequalities.
We will also briefly discuss the problem of transport on the Wiener space itself. Joint work with Dan Mikulincer.