arXiv subjects

Juliane Krautz

Publications and source records attributed to Juliane Krautz.

4 recordsLinked to original sources

arXiv2026

Dynamic Optimal Transport with Optimal Preferential Paths

We study a dynamic optimal transport type problem on a domain that consists of two parts: a compact set $Ω\subset \mathbb{R}^d$ (bulk) and a non-intersecting and sufficiently regular curve $Γ\subset Ω$. On each of them, a Benamou-Brenier type dynamic optimal transport problem is considered, yet with an additional mechanism that allows the exchange (at a cost) of mass between bulk and curve. In the respective actions, we also allow for non-linear mobilities. We first ensure the existence of minimizers by relying on the direct method of calculus of variations and we study the asymptotic properties of the minimizers under changes in the parameters regulating the dynamics in $Ω$ and $Γ$. Then, we study the case when the curve $Γ$ is also allowed to change, being the main interest in this paper. To this end, the Tangent-Point energy is added to the action functional in order to preserve the regularity properties of the curve and prevent self-intersections. Also in this case, by relying on suitable compactness estimates both for the time-dependent measures and the curve $Γ$, the existence of optimizers is shown. We extend these analytical findings by numerical simulations based on a primal-dual approach that illustrate the behaviour of geodesics, for fixed and varying curves.

math.AP↗

arXiv2026

The Schrödinger problem on metric graphs

We study the Schrödinger problem on metric graphs and its different formulations. Starting from a static version, we introduce an equivalent reformulation as entropic optimal transport and show $Γ$-convergence towards static optimal transport. We then rigorously derive a Benamou-Brenier type dynamic version of the Schrödinger problem, thereby extending known results from ${\rm RCD}^*(K,N)$-spaces. With this equivalence at hand, we conclude that the minimum values of the dynamic Schrödinger problem converge towards the squared Wasserstein distance, and minimizers converge to Wasserstein geodesics. We also extend the dynamic formulation to a more general class of initial and final data and show existence of solutions in this setting using the direct method. Lastly, we illustrate our analytical findings by a numerical investigation.

math.AP↗

arXiv2025

Weak curvature conditions on metric graphs

Starting from pointwise gradient estimates for the heat semigroup, we study three characterizations of weak lower curvature bounds on metric graphs. More precisely, we prove the equivalence between a weak notion of the Bakry-Émery curvature condition, a weak Evolutionary Variational Inequality and a weak form of geodesic convexity. The proof is based on a careful regularization of absolutely continuous curves together with an explicit representation of the Cheeger energy. We conclude with a brief discussion on possible applications to the Schrödinger bridge problem on metric graphs.

math.AP↗

arXiv2025

Dynamic Optimal Transport with optimal star shaped graphs

We study an optimal transport problem in a compact convex set $Ω\subset\mathbb{R}^d$ where bulk transport is coupled to dynamic optimal transport on a metric graph $ \mathsf{G} = (\mathsf{V},\mathsf{E})$ which is embedded in $Ω$. We prove existence of solutions for fixed graphs. Next, we consider varying graphs, yet only for the case of star-shaped ones. Here, the action functional is augmented by an additional penalty that prevents the edges of the graph to overlap. This allows to preserve the graph topology and thus to rely on standard techniques in Calculus of Variations in order to show existence of minimizers.

math.AP↗