The Schrödinger problem on metric graphs

Technical Summary: The Schrödinger Problem on Metric Graphs

Problem Statement
This paper investigates the Schrödinger problem on metric graphs, a stochastic interpolation problem concerned with finding the most likely evolution of independent gas particles between two observed distributions, $\mu_0$ and $\mu_1$. The study focuses on establishing the relationship between this problem and optimal transport (OT) within the specific geometric setting of metric graphs. Unlike Riemannian manifolds or $RCD^*(K, N)$-spaces where a comprehensive theory exists, metric graphs present unique challenges due to their lack of smoothness and the absence of strong synthetic lower curvature bounds. The authors aim to rigorously derive static and dynamic formulations of the Schrödinger problem on these spaces, prove their equivalence, and analyze their convergence to the squared Wasserstein distance as the regularization parameter vanishes.

Methodology
The authors employ a combination of analytical techniques and numerical methods:

1. Static Analysis and $\Gamma$-Convergence:

   • The static Schrödinger problem is reformulated as an entropic optimal transport (EOT) problem. The reference measure is chosen as the heat kernel $h_{\beta/2}$ on the metric graph.
   • To prove $\Gamma$-convergence of the EOT problem to the static Kantorovich optimal transport problem as the regularization parameter $\beta \to 0$, the authors utilize a "block approximation" technique adapted from [CDPS17].
   • Crucially, they leverage Gaussian upper and lower bounds for the heat kernel (derived from semigroup properties and path-sum formulas) to handle the cost function, which differs from the standard quadratic distance due to the graph structure.

2. Dynamic Formulation and Equivalence:

   • A dynamic version of the Schrödinger problem (DSP) is derived, modeled as a minimization over weak solutions to the continuity equation. The functional includes a kinetic energy term and a Fisher information term (arising from the entropy regularization).
   • The equivalence between the static (S$\beta$) and dynamic (DSP) formulations is established by constructing specific solutions using the heat flow semigroup. This involves defining time-dependent densities via the heat kernel and verifying they satisfy the continuity equation and coupling conditions on the graph vertices.
   • The authors address the lack of strong curvature bounds (which typically facilitate the construction of recovery sequences in dynamic settings) by shifting regularity requirements to the reference measure rather than the recovery sequence itself.

3. Existence and Generalization:

   • Existence of solutions for the dynamic problem is proven using the direct method of the Calculus of Variations. To handle cases where densities may not be bounded, the authors extend the functional to include singular parts of measures and utilize a modified action functional based on the convex function $\Psi(u, v) = |v|^2/2u$.
   • Compactness of the solution set is established using Hölder estimates and properties of the heat kernel on metric graphs.

4. Numerical Investigation:

   • A primal-dual algorithm based on augmented Lagrangian formulations is proposed to solve the dynamic problem. The method employs a mixed formulation introducing an auxiliary variable for the gradient of the density.
   • Discretization is performed using central difference quotients for interior points and one-sided differences at vertices, preserving the time-symmetric structure.

Key Contributions and Results

• Static Equivalence and Convergence: The paper proves that the static Schrödinger problem on metric graphs is equivalent to an entropic optimal transport problem with a modified cost function. It establishes $\Gamma$-convergence of this static entropic problem to the static Kantorovich problem (squared Wasserstein distance) as $\beta \to 0$.
• Dynamic Equivalence: The authors rigorously derive a Benamou-Brenier type dynamic formulation for the Schrödinger problem on metric graphs. They prove that for bounded initial and final data, the static and dynamic formulations are equivalent, and the dynamic problem admits a unique minimizer.
• Convergence of Minima: As a consequence of the static-dynamic equivalence and the static $\Gamma$-convergence, the minimum values of the dynamic Schrödinger problem converge to the squared Wasserstein distance, and the minimizers converge to Wasserstein geodesics.
• Handling Curvature Deficits: The paper highlights a significant theoretical distinction: while $\Gamma$-convergence holds for the static formulations, the authors note that $\Gamma$-convergence for the dynamic formulations does not follow directly on metric graphs due to the lack of strong lower curvature bounds (specifically, the absence of a sharp Wasserstein contractivity estimate without an additional constant). The convergence of minima is instead achieved via the static-dynamic equivalence.
• Existence for General Data: The dynamic formulation is extended to more general initial and final data (dropping the bounded density assumption), and the existence of solutions is proven in this broader setting.
• Numerical Validation: Numerical experiments on star-shaped graphs demonstrate the convergence of the dynamic Schrödinger solutions to the Wasserstein geodesic as $\beta \to 0$. The results illustrate the smoothing effect of the Fisher information term (continuity at vertices) for $\beta > 0$, contrasting with the discontinuous nature of standard optimal transport geodesics on graphs.

Significance
The paper provides the first comprehensive picture of the Schrödinger problem on metric graphs, filling a gap in the literature where such results were previously limited to $RCD^*(K, N)$-spaces. It demonstrates that despite the failure of strong synthetic curvature conditions on metric graphs, the fundamental relations between static and dynamic Schrödinger problems and their connection to optimal transport can be rigorously established. The work clarifies the specific role of heat kernel regularity in compensating for the lack of curvature bounds and offers a robust numerical framework for solving these problems on non-smooth metric spaces.