分析偏微分方程的新几何框架
A new Geometric Setting for the Analysis of Partial Differential Equations
AI总结：
本文结合Wasserstein与Aitchison几何提出混合度量$D_\alpha$，证明其性质并建立梯度流理论，将若干偏微分方程作为其梯度流，还与Wasserstein--Fisher--Rao度量对比。
AI中文摘要：
本文在绝对连续概率密度空间上引入一种混合度量几何，结合最优传输（Wasserstein几何）与对数比合成（Aitchison几何）。混合距离$D_\alpha$通过Benamou--Brenier型动力学公式定义，该公式将空间传输与保持总量的中心化反应项耦合。我们证明$D_\alpha$是真正的度量，并建立其与Wasserstein距离、Aitchison距离的比较估计。特别地，我们表明$D_\alpha$诱导的拓扑比窄拓扑更强，比Wasserstein与Aitchison度量生成的上确界拓扑更弱。我们进一步证明该度量空间是测地线空间。在该框架内，我们发展了Ambrosio--Gigli--Savaré意义下的梯度流理论基础，包括绝对连续曲线的刻画、度量导数、度量斜率及形式Jordan--Kinderlehrer--Otto格式。我们还研究了混合重心及其与Wasserstein重心、Aitchison重心的联系。最后，我们讨论了若干偏微分方程，包括逻辑扩散、带对数比约束的Allen--Cahn方程、带对数增长的趋化模型，作为与该混合几何相关的形式梯度流，并将所提框架与Wasserstein--Fisher--Rao度量进行比较。
英文摘要：
In this paper, we introduce a hybrid metric geometry on the space of absolutely continuous probability densities that combines optimal transport (Wasserstein geometry) and log-ratio composition (Aitchison geometry). The hybrid distance $D_α$ is defined through a Benamou--Brenier-type dynamical formulation that couples spatial transport with a centered reaction term preserving total mass.We prove that $D_α$ is a genuine metric and establish comparison estimates with the Wasserstein and Aitchison distances. In particular, we show that the topology induced by $D_α$ is stronger than the narrow topology and weaker than the supremum topology generated by the Wasserstein and Aitchison metrics. We further prove that the metric space is geodesic. Within this framework, we develop the foundations of a gradient flow theory in the sense of Ambrosio--Gigli--Savaré, including the characterization of absolutely continuous curves, metric derivatives, metric slopes, and formal Jordan--Kinderlehrer--Otto schemes. We also investigate hybrid barycenters and their connections with Wasserstein barycenters and Aitchison barycenters. Finally, we discuss several partial differential equations, including logistic diffusion, Allen--Cahn equations with log-ratio constraints, and chemotaxis models with logarithmic growth, as formal gradient flows associated with the hybrid geometry, and compare the proposed framework with the Wasserstein--Fisher--Rao metric.
