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.

中文翻译：

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$${\textsf {RCD}}^*(K,N)$$ RCD ∗ ( K , N ) 空间熵成本的 Benamou–Brenier 和对偶公式

在本文中，我们证明，在 $$\textsf {RCD}^*(K,N)$$RCD∗(K,N) 空间和 $$N<\infty $$N<∞ 的框架内，熵cost（即薛定谔问题的最小值）承认：三重动态变分表示，本着Wasserstein距离的Benamou-Brenier公式的精神；Hamilton-Jacobi-Bellman对偶表示，符合Bobkov-Gentil-Ledoux和奥托-维拉尼 (Otto-Villani) 得出了哈密顿-雅可比 (Hamilton-Jacobi) 与最优输运的连续性方程之间的对偶性；Kantorovich 型对偶性公式，其中 Hopf-Lax 半群被合适的“熵”对应物取代。因此，我们提供了薛定谔问题的等效变分表示的完整和统一图，以及与 Wasserstein 距离的类似公式的完美并行性。