We consider an optimal transport problem on the unit simplex whose solutions are given by gradients of exponentially concave functions and prove two main results. First, we show that the optimal transport is the large deviation limit of a particle system of Dirichlet processes transporting one probability measure on the unit simplex to another by coordinatewise multiplication and normalizing. The structure of our Lagrangian and the appearance of the Dirichlet process relate our problem closely to the entropic measure on the Wasserstein space as defined by von-Renesse and Sturm in the context of Wasserstein diffusion. The limiting procedure is a triangular limit where we allow simultaneously the number of particles to grow to infinity while the `noise' tends to zero. The method, which generalizes easily to many other cost functions, including the squared Euclidean distance, provides a novel combination of the Schrodinger problem approach due to C. Leonard and the related Brownian particle systems by Adams et al.which does not require gamma convergence. Second, we analyze the behavior of entropy along the paths of transport. The reference measure on the simplex is taken to be the Dirichlet measure with all zero parameters which relates to the finite-dimensional distributions of the entropic measure. The interpolating curves are not the usual McCann lines. Nevertheless we show that entropy plus a multiple of the transport cost remains convex, which is reminiscent of the semiconvexity of entropy along lines of McCann interpolations in negative curvature spaces. We also obtain, under suitable conditions, dimension-free bounds of the optimal transport cost in terms of entropy.

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乘法薛定谔问题和狄利克雷传输

我们考虑了单位单纯形上的最优传输问题，其解由指数凹函数的梯度给出，并证明了两个主要结果。首先，我们证明了最优传输是 Dirichlet 过程的粒子系统的大偏差极限，通过坐标乘法和归一化将单位单纯形上的一个概率度量传输到另一个。我们的拉格朗日函数的结构和狄利克雷过程的出现将我们的问题与 von-Renesse 和 Sturm 在 Wasserstein 扩散的上下文中定义的 Wasserstein 空间的熵测度密切相关。限制程序是一个三角形限制，我们同时允许粒子数增长到无穷大，而“噪声”趋于零。该方法很容易推广到许多其他成本函数，包括平方欧几里得距离，提供了一种新的薛定谔问题方法的组合，这是由于 C. Leonard 和 Adams 等人的相关布朗粒子系统，不需要伽马收敛。其次，我们分析了熵沿传输路径的行为。单纯形上的参考测度被认为是具有所有零参数的狄利克雷测度，这与熵测度的有限维分布有关。插值曲线不是通常的麦肯线。尽管如此，我们表明熵加上运输成本的倍数仍然是凸的，这让人想起负曲率空间中沿着麦肯插值线的熵的半凸性。在合适的条件下，我们还获得了在熵方面的最优运输成本的无量纲边界。