A new Geometric Setting for the Analysis of Partial Differential Equations

Abstract

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 $D_\alpha$ is defined through a Benamou--Brenier-type dynamical formulation that couples spatial transport with a centered reaction term preserving total mass.We prove that $D_\alpha$ is a genuine metric and establish comparison estimates with the Wasserstein and Aitchison distances. In particular, we show that the topology induced by $D_\alpha$ 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--Savar\'e, 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.