A special formulation of the Schrödinger problem consists in minimizing some action on the Wasserstein space of probability measures on a Riemannian manifold subject to fixed initial and final data. We extend this action minimization problem by replacing the usual entropy, underlying the Schrödinger problem, with a general function on the Wasserstein space. The corresponding minimal cost approaches the squared Wasserstein distance when the fluctuation parameter $\varepsilon$ tends to zero.

We show heuristically that the solutions satisfy some Newton equation, extending a recent result of Conforti. The connection with Wasserstein gradient flows is established and various inequalities, including evolutional variational inequalities and contraction inequalities under a curvature-dimension condition, are derived with a heuristic point of view. As a rigorous result we prove a new and general contraction inequality for the Schrödinger problem under a Ricci lower bound on a smooth and compact Riemannian manifold.

Schrödinger问题的一种特殊表示形式是，在固定初始和最终数据的情况下，使对黎曼流形上的概率测度的Wasserstein空间的某些操作最小化。我们通过用Wasserstein空间上的一般函数替换Schrödinger问题基础上的通常的熵来扩展该动作最小化问题。当波动参数$ \ varepsilon $趋于零时，相应的最小成本接近Wasserstein距离的平方。

我们试探性地表明，这些解满足某些牛顿方程，从而扩展了Conforti的最新结果。建立了与Wasserstein梯度流的联系，并从启发式的角度推导了各种不等式，包括曲率维条件下的演化变分不等式和收缩不等式。作为严格的结果，我们证明了在光滑紧致黎曼流形上的Ricci下界下的Schrödinger问题的一个新的广义压缩不等式。