Diffusive transport on the real line: semi-contractive gradient flows and their discretization
Daniel Matthes, Eva-Maria Rott, André Schlichting
TL;DR
The paper introduces the diffusive transport distance $\mathbb{D}$ as a diffusion-constrained pseudo-distance between probability measures on $\mathbb{R}$, linking it to the Hellinger and Kantorovich metrics and enabling a gradient-flow interpretation for fourth-order PDEs. It establishes a formal gradient-flow theory showing semi-contractive behavior for energy functionals under $\mathbb{D}$ and then develops a rigorous spatial discretization on $\delta\mathbb{Z}$ that preserves contractivity with rates uniform in the lattice spacing. The discrete theory defines appropriate difference operators, function spaces, and a discrete optimal diffusive connection, proving the existence of geodesics and curves of finite action, and deriving mesh-uniform contraction results for external potentials, convolution-based interactions, and the quadratic porous medium equation. These results provide a structure-preserving numerical framework for higher-order diffusion-type equations and illuminate the relation between diffusive and Wasserstein-type gradient flows, with potential applications in numerical analysis and kinetic models.
Abstract
The diffusive transport distance, a novel pseudo-metric between probability measures on the real line, is introduced. It generalizes Martingale optimal transport, and forms a hierarchy with the Hellinger and the Wasserstein metrics. We observe that certain classes of parabolic PDEs, among them the porous medium equation of exponent two, are formally semi-contractive metric gradient flows in the new distance. This observation is made rigorous for a suitable spatial discretization of the considered PDEs: these are semi-contractive gradient flows with respect to an adapted diffusive transport distance for measures on the point lattice. The main result is that the modulus of convexity is uniform with respect to the lattice spacing. Particularly for the quadratic porous medium equation, this is in contrast to what has been observed for discretizations of the Wasserstein gradient flow structure.