论文标题
纤维纤维最佳运输拓扑中的异质梯度流
Heterogeneous gradient flows in the topology of fibered optimal transport
论文作者
论文摘要
我们在纤维束的概率度量方面引入了最佳的运输拓扑，这对从一个纤维到另一个纤维的运输成本进行了惩罚。为简单起见，我们在Euclidean Case $ \ Mathbb {r}^d \ times \ Mathbb {r}^d $中说明了我们的构造，在其中我们对第二部分中的二次成本进行了惩罚。然后，最佳运输被限制在固定纤维沿着固定纤维沿岸发生。尽管无限价值和不连续的成本脱节，但我们证明概率空间测量$（\ Mathcal {p} _ {2，ν}（\ Mathbb {r}^{2d {2d}），W_ {2，ν}） \ Mathcal {p}（\ Mathbb {r}^d）$在第二个组件中变成了纤维运输距离下的波兰空间，在光纤的传输距离下，它具有弱的利曼尼亚结构，让人联想到F. otto为经典的Quadratic Quadratic Wasserstein空间提出的一种。解决了三个基本问题：1）关于新拓扑的梯度流的抽象理论； 2）我们展示了在具有异质性的大量进化PDE上识别新型的纤维梯度流结构的应用； 3）我们利用我们的方法在具有弱奇异耦合的多维cucker-smale型比对模型中得出长期行为和全球平均场限制。
We introduce an optimal transport topology on the space of probability measures over a fiber bundle, which penalizes the transport cost from one fiber to another. For simplicity, we illustrate our construction in the Euclidean case $\mathbb{R}^d\times \mathbb{R}^d$, where we penalize the quadratic cost in the second component. Optimal transport becomes then constrained to happen along fixed fibers. Despite the degeneracy of the infinitely-valued and discontinuous cost, we prove that the space of probability measures $(\mathcal{P}_{2,ν}(\mathbb{R}^{2d}),W_{2,ν})$ with fixed marginal $ν\in \mathcal{P}(\mathbb{R}^d)$ in the second component becomes a Polish space under the fibered transport distance, which enjoys a weak Riemannian structure reminiscent of the one proposed by F. Otto for the classical quadratic Wasserstein space. Three fundamental issues are addressed: 1) We develop an abstract theory of gradient flows with respect to the new topology; 2) We show applications that identify a novel fibered gradient flow structure on a large class of evolution PDEs with heterogeneities; 3) We exploit our method to derive long-time behavior and global-in-time mean-field limits in a multidimensional Cucker-Smale-type alignment model with weakly singular coupling.
