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We provide a solution to the problem of optimal transport by Brownianmartingales in general dimensions whenever the transport cost satisfies certainsubharmonic properties in the target variable, as well as a stochastic versionof the standard "twist condition" frequently used in deterministic Mongetransport theory. This setting includes in particular the case of the distancecost $c(x,y)=|x-y|$. We prove existence and uniqueness of the solution andcharacterize it as the first time Brownian motion hits a barrier that isdetermined by solutions to a corresponding dual problem.

A solution to the Monge transport problem for Brownian martingales