纤维化最优传输拓扑中的异质梯度流
Heterogeneous gradient flows in the topology of fibered optimal transport
AI总结：
本文在纤维丛概率测度空间引入惩罚跨纤维传输成本的最优传输拓扑，证明其构成具有弱黎曼结构的波兰空间，发展相关梯度流理论，并应用于演化PDE及Cucker-Smale模型的长时行为与平均场极限推导。
摘要：
我们在纤维丛上的概率测度空间中引入了一种最优传输拓扑，该拓扑对从一个纤维到另一个纤维的传输成本进行惩罚。为简便起见，我们在欧几里得情形 $\mathbb{R}^d\times \mathbb{R}^d$ 中说明我们的构造，其中我们对第二个分量中的二次成本进行惩罚。于是，最优传输被限制为沿着固定纤维发生。尽管无穷值且不连续的成本存在退化性，我们证明了在第二个分量中具有固定边缘 $ν\in \mathcal{P}(\mathbb{R}^d)$ 的概率测度空间 $(\mathcal{P}_{2,ν}(\mathbb{R}^{2d}),W_{2,ν})$ 在纤维化传输距离下成为一个波兰空间，该空间具有弱黎曼结构，让人联想到 F. Otto 为经典二次 Wasserstein 空间提出的结构。我们探讨了三个基本问题：1）我们发展了关于新拓扑的梯度流抽象理论；2）我们展示了在具有异质性的一大类演化 PDE 上识别新型纤维化梯度流结构的应用；3）我们利用我们的方法，推导了具有弱奇异耦合的多维 Cucker-Smale 型一致模型中的长时间行为和全局时间平均场极限。
英文摘要：
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.
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