Benamou–Brenier and duality formulas for the entropic cost on $${\textsf {RCD}}^*(K,N)$$RCD∗(K,N) spaces
Benamou–Brenier and duality formulas for the entropic cost on $${\textsf {RCD}}^*(K,N)$$RCD∗(K,N) spaces
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$${ extsf {RCD}}^*(K,N)$$RCD∗(K,N) 空间上熵成本的 Benamou–Brenier 和对偶公式
DOI：
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发表时间：
2018
影响因子：
2
通讯作者：
Luca Tamanini
中科院分区：
文献类型：
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作者：
N. Gigli;Luca Tamanini
在本文中，我们证明了在具有∗(K，N)和∞(K，N)的Textsf{rcd}^*(K，N)空间的框架内，熵代价(即薛定谔问题的最小值)允许：一个三重动力变分表示，其精神是关于Wasserstein距离的Benamou-Brenier公式；一个Hamilton-Jacobi-Bellman对偶表示，符合Bobkov-Gentil-Ledoux和Otto-Villani关于最优运输的Hamilton-Jacobi和连续性方程的对偶结果；一个Kantorovich类型的对偶公式，其中Hopf-Lax半群被一个合适的‘熵’对偶取代。因此，我们给出了薛定谔问题的等价变分表示的完整而统一的图解，并与Wasserstein距离的类似公式完美地平行。Ricci曲率下有界的黎曼流形是一类相关的$$\Textsf{Rcd}^*(K，N)$$Rcd∗(K，N)空间，即使在这种情况下，我们的结果也是新的。
In this paper we prove that, within the framework of $$\textsf {RCD}^*(K,N)$$RCD∗(K,N) spaces with $$N<\infty $$N<∞, the entropic cost (i.e. the minimal value of the Schrödinger problem) admits:A threefold dynamical variational representation, in the spirit of the Benamou–Brenier formula for the Wasserstein distance;A Hamilton–Jacobi–Bellman dual representation, in line with Bobkov–Gentil–Ledoux and Otto–Villani results on the duality between Hamilton–Jacobi and continuity equation for optimal transport;A Kantorovich-type duality formula, where the Hopf–Lax semigroup is replaced by a suitable ‘entropic’ counterpart. We thus provide a complete and unifying picture of the equivalent variational representations of the Schrödinger problem as well as a perfect parallelism with the analogous formulas for the Wasserstein distance. Riemannian manifolds with Ricci curvature bounded from below are a relevant class of $$\textsf {RCD}^*(K,N)$$RCD∗(K,N) spaces and our results are new even in this setting.