论文标题
有限添加的质量运输
Finitely additive mass transportation
论文作者
论文摘要
在有限的附加环境中研究了一些经典的大规模运输问题。令$ω= \ prod_ {i = 1}^nΩ_i$和$ \ nathcal {a} = \ outimes_ {i = 1}^n \ nathcal {a} _i $，其中$（ω_i，\ mathcal，\ nathcal {a} _i} _i，μ_i） $ i = 1，\ ldots，n $。令$ c：ω\ rightarrow [0，\ infty] $为$ \ mathcal {a} $ - 可测量的成本函数。令$ m $为$ \ Mathcal {a} $的有限添加概率的集合，带有边际$μ_1，\ ldots，μ_n$。如果将耦合视为$ m $的要素，那么大多数大众运输理论的经典结果（包括二元性和坎托罗维奇INF的可实现性）是有效的，没有任何进一步的假设。特别注意Martingale运输。令所有$ i $和$ i $ and $ i $ and $ i $ and $ m _1 = \ bigl \ {p \ bigl \ {p \ bigl \ {p \ ll p \ ll p^*\ p \ ll p^*\ p \ bigl \ { }（π_1，\ ldots，π_n）\ text {是a} p \ text {-martingale} \} $$，其中$ p^*$是$ \ mathcal {a a} $的参考概率。如果$ m_1 \ ne \ emptySet $，则$$ \ int c \，dp = \ inf_ {q \ in m_1}} \ int c \，dq \ quad \ quad \ quad \ quad \ quad \ text {for some} p \ for M_1。
Some classical mass transportation problems are investigated in a finitely additive setting. Let $Ω=\prod_{i=1}^nΩ_i$ and $\mathcal{A}=\otimes_{i=1}^n\mathcal{A}_i$, where $(Ω_i,\mathcal{A}_i,μ_i)$ is a ($σ$-additive) probability space for $i=1,\ldots,n$. Let $c:Ω\rightarrow [0,\infty]$ be an $\mathcal{A}$-measurable cost function. Let $M$ be the collection of finitely additive probabilities on $\mathcal{A}$ with marginals $μ_1,\ldots,μ_n$. If couplings are meant as elements of $M$, most classical results of mass transportation theory, including duality and attainability of the Kantorovich inf, are valid without any further assumptions. Special attention is devoted to martingale transport. Let $(Ω_i,\mathcal{A}_i)=(\mathbb{R},\mathcal{B}(\mathbb{R}))$ for all $i$ and $$M_1=\bigl\{P\in M:P\ll P^*\text{ and }(π_1,\ldots,π_n)\text{ is a }P\text{-martingale}\}$$ where $P^*$ is a reference probability on $\mathcal{A}$. If $M_1\ne\emptyset$, then $$\int c\,dP=\inf_{Q\in M_1}\int c\,dQ\quad\quad\text{for some }P\in M_1.$$ Conditions for $M_1\ne\emptyset$ are given as well.
