  We provide a solution to the problem of optimal transport by Brownian
martingales in general dimensions whenever the transport cost satisfies certain
subharmonic properties in the target variable, as well as a stochastic version
of the standard "twist condition" frequently used in deterministic Monge
transport theory. This setting includes in particular the case of the distance
cost $c(x,y)=|x-y|$. We prove existence and uniqueness of the solution and
characterize it as the first time Brownian motion hits a barrier that is
determined by solutions to a corresponding dual problem.
