One of the central objects in the theory of optimal transport is the Brenier map: the unique monotone transformation which pushes forward an absolutely continuous probability law onto any other given law. A line of recent work has analyzed $L^2$ convergence rates of plugin estimators of Brenier maps, which are defined as the Brenier map between density estimators of the underlying distributions. In this work, we show that such estimators satisfy a pointwise central limit theorem when the underlying laws are supported on the flat torus of dimension $d \geq 3$. We also derive a negative result, showing that these estimators do not converge weakly in $L^2$ when the dimension is sufficiently large. Our proofs hinge upon a quantitative linearization of the Monge-Amp\`ere equation, which may be of independent interest. This result allows us to reduce our problem to that of deriving limit laws for the solution of a uniformly elliptic partial differential equation with a stochastic right-hand side, subject to periodic boundary conditions.

翻译：最优传输理论的核心对象之一是Brenier映射：将绝对连续概率律推前到任意给定概率律的唯一单调变换。近期一系列工作分析了Brenier映射插件估计量的$L^2$收敛速率，该估计量定义为底层分布密度估计量之间的Brenier映射。本文证明，当底层律支撑在维度$d \geq 3$的平坦环面上时，此类估计量满足逐点中心极限定理。我们还推导出一个否定性结果：当维度足够大时，这些估计量在$L^2$中不弱收敛。我们的证明依赖于Monge-Ampère方程的定量线性化，该结果本身可能具有独立意义。这一结果使我们能够将问题简化为推导具有随机右端项且满足周期边界条件的一致椭圆型偏微分方程解的极限定律。