本文主要研究Wasserstein空间W₂中的Fréchet均值。基于国内外目前的相关成果，先简要介绍最优运输问题及Wasserstein空间，然后在空间W₂(X)中展开分析。针对“X为可分的Hilbert空间”这一特殊情形，提炼出Fréchet均值的存在性、唯一性、连续性等核心性质，对比、整理各研究角度的异同点，并总结出必要的、简洁的证明思路。利用多重耦合推导了经验Fréchet均值的存在性和连续性，而唯一性的论述则主要借助凸分析的工具。在探究经验Fréchet泛函的可微性时，引入了Karcher均值的概念并阐明其与经验Fréchet均值的联系，介绍了使二者相等的正则性条件。类似地，定义了对应于随机测度的总体Fréchet均值，结合分析学，着重从概率与统计的角度展开讨论。以X=R^d的情况为例证明存在性，然后简述更一般的结论；给出唯一性的关键推理步骤，并集中叙述其他重要性质和命题。

This paper mainly studies Fréchet means in the Wasserstein space W₂. Based on relevant literature, it firstly makes a brief introduction to the optimal transport problem and the Wasserstein space, and then analyzes Fréchet means in the space W₂(X). In view of the special case that X is a separable Hilbert space, the article highlights some important properties of
Fréchet means, such as the existence, uniqueness and continuity. By comparing various methods, it summarizes concise proof ideas. The existence and continuity of empirical Fréchet means are derived by using multicouplings, while the uniqueness is mainly discussed by means of convex analysis. In the study of differentiability of the Fréchet functional, the concept of the Karcher mean is introduced, and the regularity conditions that make it equal the Fréchet mean are given. The population Fréchet mean of a random measure is defined in a similar way, and then the discussion from the perspective of probability and statistics is made combining with the knowledge of analysis. The case of X=R^d is taken as an example to prove the existence of population Fréchet means, followed by a more general conclusion. The key steps of
inferring uniqueness are given and other important properties and propositions are described.