关于R上Wasserstein空间中的绝对连续曲线及其通过最优马尔可夫过程的表示
On absolutely continuous curves in the Wasserstein space over R and their representation by an optimal Markov process
AI总结：
本文研究R上Wasserstein空间中绝对连续曲线，通过最优马尔可夫过程（马尔可夫分位数过程）给出其唯一的最小拉格朗日概率代表，并探讨获得此类代表及唯一性的新途径。
摘要：
设$\mu$ = ($\mu$t)t$\in$R是R上的一族单参数概率测度。在文献[11]中，我们引入了其“马尔可夫分位数”过程：一个过程X= (Xt)t$\in$R，在附着于$\mu$的马尔可夫过程中，即其边际律族为$\mu$的过程中，尽可能类似于附着于$\mu$的分位数过程。本文研究$\mu$在Wasserstein空间P2(R)中绝对连续的情形。此时X是具有边际($\mu$t)t的动力输运问题的解。它提供了$\mu$的一个马尔可夫最小拉格朗日概率代表，并且该代表在通过某些类型的极限获得的进程中具有唯一性：这些极限包括有限维拓扑下分位数过程的极限（其中在过去与未来在给定现在条件下于有限多个时刻独立），或线性插值$\mu$的过程的极限。这提出了关于获得马尔可夫拉格朗日代表的方式以及在此框架中寻求唯一性性质的新问题。
英文摘要：
Let $μ$ = ($μ$t)t$\in$R be a 1-parameter family of probability measures on R. In [11] we introduced its ``Markov-quantile''process: a process X= (Xt)t$\in$R that resembles as much as possible the quantile process attached to $μ$, among the Markov processesattached to $μ$, i.e. whose family of marginal laws is $μ$.In this article we look at the case where $μ$ is absolutely continuous in the Wasserstein space P2(R). Then X is solution of adynamical transport problem with marginals ($μ$t)t. It provides a Markov minimal Lagrangian probabilistic representative of $μ$, whichis moreover unique among the processes obtained as certain types of limits: limits for the finite dimensional topology of quantileprocesses where the past is made independent of the future conditionally on the present at finitely many times, or limits of processeslinearly interpolating $μ$.This raises new questions about ways to obtain Markov Lagrangian representatives, and to seek uniqueness properties in thisframework.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
