论文标题
将熵在最佳运输和计划问题中的效果正规化效果
Regularizing effects of the entropy functional in optimal transport and planning problems
论文作者
论文摘要
我们分析了最佳的运输问题，并在Wasserstein空间中沿曲线评估的其他熵成本，该成本加入了两个概率指标$ M_0，M_1 $。附加的熵功能结果对双重问题的（所谓的）Kantorovich电位的椭圆正则化。假设初始和终端度量为正和平滑，我们证明了最佳曲线一直保持平稳。我们重点介绍了在$ d $维欧几里得空间（边界上没有升华条件）中设置在凸有界域上的传输问题的情况，但是我们还提到了整个空间中类似高斯的测量的情况。该方法遵循P.-L。介绍的想法。平均场游戏理论中的狮子\ cite {l-college}。该结果提供了在惩罚术语中的优化问题中平稳近似，这些问题出现在平均场控制或平均场计划问题中。这使我们能够通过使用Eulerian方法中的位移凸性属性来利用这种问题的新估计。
We analyze optimal transport problems with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures $m_0,m_1$. The effect of the additional entropy functional results into an elliptic regularization for the (so-called) Kantorovich potentials of the dual problem. Assuming the initial and terminal measures to be positive and smooth, we prove that the optimal curve remains smooth for all time. We focus on the case that the transport problem is set on a convex bounded domain in the $d$-dimensional Euclidean space (with no-flux condition on the boundary), but we also mention the case of Gaussian-like measures in the whole space. The approach follows ideas introduced by P.-L. Lions in the theory of mean-field games \cite{L-college}. The result provides with a smooth approximation of minimizers in optimization problems with penalizing congestion terms, which appear in mean-field control or mean-field planning problems. This allows us to exploit new estimates for this kind of problems by using displacement convexity properties in the Eulerian approach.
