Development of geometric analysis based on coupling methods for diffusion processes

在不允许可微结构的奇异空间上，我们将热流的两种不同形式标识为梯度流。利用随机分析技巧，我们将热分布之间的Wasserstein距离的一个估计推广到了关于Ricci流的一个估计.此外，我们将其推广到由Perelman的L距离定义的类Wasserstein型距离，得到了镜面耦合的最优输运的特征性质的公式。利用这个公式，我们证明了由黎曼流形的极限给出的一类奇异空间也具有相同的性质。

On singular spaces admitting no differentiable structure, we identify two different formulations of heat flows as gradient flows. As a result, we prove an implication from a formulation of the presence of a lower Ricci curvature bound via optimal transport to another one via the gradient estimate of heat semigroups.By using stochastic analytic techniques, we extend an estimate of the Wasserstein distance between heat distributions which appears as a kind of curvature condition in the above argument to the one on Ricci flows. In addition, we extend it to the one associated with the Wasserstein-like distance defined by using Perelman's L-distance.We obtain a formulation of a characteristic property of the mirror coupling in terms of optimal transport. By using this formulation, we proved that the same property holds on a class of singular spaces given by limit of Riemannian manifolds.