# A solution to the Monge transport problem for Brownian martingales

## TL;DR

This paper solves the optimal transport problem for Brownian martingales in multiple dimensions under specific cost and subharmonic conditions, establishing existence, uniqueness, and a barrier-based characterization of the solution.

## Contribution

It introduces a novel solution framework for the Monge transport problem involving Brownian martingales with subharmonic costs, including the distance cost case.

## Key findings

- Proves existence and uniqueness of the optimal transport solution.
- Characterizes the solution as the first hitting time of a barrier by Brownian motion.
- Extends the theory to general dimensions and specific cost functions.

## Abstract

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.