最优输运和规划问题中熵泛函的正则化效应
Regularizing effects of the entropy functional in optimal transport and planning problems
摘要：
我们分析了最优输运问题的额外的熵成本评估沿曲线在Wasserstein空间加入两个概率测度$m_0,m_1$。额外的熵泛函的效果的结果到一个椭圆正则化的（所谓的）Kantorovich潜在的对偶问题。假设初始和终端措施是积极的和光滑的，我们证明了最佳曲线保持光滑的所有时间。我们主要讨论了在$d$维欧几里得空间中的凸有界区域上的迁移问题（边界上无通量条件），同时也讨论了整个空间中的Gaussian-like测度.该方法遵循P. - L.狮子在平均场博弈理论\cite{L-college}。结果提供了一个光滑的近似极小值的最优化问题与惩罚拥塞条款，出现在平均场控制或平均场规划问题。这使我们能够利用新的估计，这类问题，通过使用位移凸性质的欧拉方法。
英文摘要：
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.
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