零温下沿薛定谔桥图的熵曲率
Entropic curvature on graphs along Schr{ö}dinger bridges at zero temperature
摘要：
将测地空间上的Lott-Sturm-Villani曲率理论推广到离散图空间。L{é}onard通过在熵曲率的定义中用Schr{ö}odinger桥代替W2-Wasserstein测地线[23，25，24]。作为一个值得注意的事实，当温度参数趋于零时，这些薛定谔桥被空间的测地线所支持。我们在离散图上分析这个性质，以达到离散空间上的熵曲率。我们的方法提供了下界的熵曲率的几个例子的图形空间：格Z n赋予计数措施，离散立方体赋予产品概率测度，圆，完全图，Bernoulli-Laplace模型。我们的一般结果也适用于一个大类的图，这在本文中没有专门研究。与Erbar-Maas关于图的结果[27，10，11]不同，本文的熵曲率结果在离散空间上给出了新的Pr{é}kopa-Leindler型不等式，并在上述图上给出了与精细浓度性质有关的新的输运熵不等式.例如在离散超立方体{0，1} n上，对于Bernoulli Laplace模型，得到了一个新的W2 - W1输运熵不等式，它不能用通常的归纳法在n维上导出.作为一个令人惊讶的事实，我们的方法也给出了与Talagrand [38]的所谓凸壳方法相关的弱输运熵不等式（见[28，15]）的改进。
英文摘要：
Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. L{é}onard by replacing W2-Wasserstein geodesics by Schr{ö}odinger bridges in the definition of entropic curvature [23, 25, 24]. As a remarkable fact, as a temperature parameter goes to zero, these Schr{ö}dinger bridges are supported by geodesics of the space. We analyse this property on discrete graphs to reach entropic curvature on discrete spaces. Our approach provides lower bounds for the entropic curvature for several examples of graph spaces: the lattice Z n endowed with the counting measure, the discrete cube endowed with product probability measures, the circle, the complete graph, the Bernoulli-Laplace model. Our general results also apply to a large class of graphs which are not specifically studied in this paper. As opposed to Erbar-Maas results on graphs [27, 10, 11], entropic curvature results of this paper imply new Pr{é}kopa-Leindler type of inequalities on discrete spaces, and new transport-entropy inequalities related to refined concentration properties for the graphs mentioned above. For example on the discrete hypercube {0, 1} n and for the Bernoulli Laplace model, a new W2 -- W1 transport-entropy inequality is reached, that can not be derived by usual induction arguments over the dimension n. As a surprising fact, our method also gives improvements of weak transport-entropy inequalities (see [28, 15]) associated to the so-called convex-hull method by Talagrand [38].
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