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A weak transport approach to the Schrödinger-Bass bridge

Manuel Hasenbichler, Gudmund Pammer, Stefan Thonhauser

Abstract

We study the Schrödinger-Bass problem, a one-parameter family of semimartingale optimal transport problems indexed by $β>0$, whose limiting regimes interpolate between the classical Schrödinger bridge, the Brenier-Strassen problem, and, after rescaling, the martingale Benamou-Brenier (Bass) problem. Our first main result is a static formulation. For each $β>0$, we prove that the dynamic Schrödinger-Bass problem is equivalent to a static weak optimal transport (WOT) problem with explicit cost $C_{\mathrm{SB}}^β$. This yields primal and dual attainment, as well as a structural characterization of the optimal semimartingales, through the general WOT framework. The cost $C_{\mathrm{SB}}^β$ is constructed via an infimal convolution and deconvolution of the Schrödinger cost with the Wasserstein distance. In a broader setting, we show that such infimal convolutions preserve the WOT structure and inherit continuity, coercivity, and stability of both values and optimizers with respect to the marginals. Building on this formulation, we propose a Sinkhorn-type algorithm for numerical computation. We establish monotone improvement of the dual objective and, under suitable integrability assumptions on the marginals, convergence of the iteration to the unique optimizer. We then study the asymptotic regimes $β\uparrow\infty$ and $β\downarrow0$. We prove that the costs $C_{\mathrm{SB}}^β$ converge pointwise to the Schrödinger cost and, after natural rescaling, to the Brenier-Strassen and Bass costs. The associated values and optimal solutions are shown to converge to those of the corresponding limiting problems.

A weak transport approach to the Schrödinger-Bass bridge

Abstract

We study the Schrödinger-Bass problem, a one-parameter family of semimartingale optimal transport problems indexed by , whose limiting regimes interpolate between the classical Schrödinger bridge, the Brenier-Strassen problem, and, after rescaling, the martingale Benamou-Brenier (Bass) problem. Our first main result is a static formulation. For each , we prove that the dynamic Schrödinger-Bass problem is equivalent to a static weak optimal transport (WOT) problem with explicit cost . This yields primal and dual attainment, as well as a structural characterization of the optimal semimartingales, through the general WOT framework. The cost is constructed via an infimal convolution and deconvolution of the Schrödinger cost with the Wasserstein distance. In a broader setting, we show that such infimal convolutions preserve the WOT structure and inherit continuity, coercivity, and stability of both values and optimizers with respect to the marginals. Building on this formulation, we propose a Sinkhorn-type algorithm for numerical computation. We establish monotone improvement of the dual objective and, under suitable integrability assumptions on the marginals, convergence of the iteration to the unique optimizer. We then study the asymptotic regimes and . We prove that the costs converge pointwise to the Schrödinger cost and, after natural rescaling, to the Brenier-Strassen and Bass costs. The associated values and optimal solutions are shown to converge to those of the corresponding limiting problems.

Paper Structure

This paper contains 21 sections, 26 theorems, 300 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Schrödinger problem
  3. Weak optimal transport
  4. Martingale optimal transport
  5. The Schrödinger--Bass problem
  6. Organization of the paper
  7. Notation
  8. Main results
  9. Infimal convolution of weak transport problems
  10. Regularity assumptions
  11. Structural properties and duality
  12. Stability
  13. Fundamental theorem
  14. Deconvolution of WOT problems
  15. Applications of \ref{['thm:fundamental_inf_conv', 'prop:deconvolution']}
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $\beta>0$ and let $\mu,\nu\in\mathcal{P}_2(\mathbb R^d)$. Then where the cost $C_{\rm SB}^\beta\colon \mathbb R^d\times\mathcal{P}_2(\mathbb R^d)\to\mathbb R$ is a continuous standard weak transport cost and satisfies a quadratic growth bound. Moreover, The weak transport problem in eq:main_sbs_static_primal admits a unique optimizer, and the dynamic problem is attained by a semimartingale t

Figures (1)

  • Figure 1: Schematic illustration of the relations characterizing the Schrödinger--Bass system, where $v = q_1-\tfrac{1}{\beta}f$, $u = q_1 - \tfrac{1}{\beta} q_\beta \Box \bigl(-\mathcal{T}_\beta[f]\bigr)$. The drift and diffusion coefficients are defined by $a_t := \nabla \log(g_{Y_0,t}(Y_t))$ and $b_t := I_d + \frac{1}{\beta} \nabla^2 \log(g_{Y_0,t}(Y_t))$, respectively.

Theorems & Definitions (62)

  • Theorem 2.1: Structure of the Schrödinger--Bass problem
  • Theorem 2.2: Convergence of the Schrödinger--Bass algorithm
  • Definition 3.1: Standard weak transport costs
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4: Effective domain of a weak transport cost
  • Definition 3.5: Coercivity, continuity, and growth assumptions
  • Definition 3.6: $p$-moment control
  • Proposition 3.7: Properties of the infimal convolution
  • proof
  • ...and 52 more