Schrödinger–Bass Bridge (SBB) Framework

• SBB is a one-parameter family of semimartingale optimal transport that unifies Schrödinger bridge, Brenier–Strassen, and Bass problems by penalizing drift and volatility deviations.
• It features an analytic structure with a static weak optimal transport reformulation, leveraging strong duality and scalable Sinkhorn-type algorithms.
• Under various parameter regimes, SBB converges to distinct optimal transport limits, advancing practical applications in generative modeling and stochastic control.

The Schrödinger–Bass Bridge (SBB) is a one-parameter family of semimartingale optimal transport problems that unifies and interpolates between three central objects in probabilistic optimal transport: the classical Schrödinger bridge, the Brenier–Strassen (weak quadratic transport), and the martingale Benamou–Brenier (Bass) problems. The SBB framework penalizes both drift and volatility deviation in a continuous-time Markovian coupling between prescribed initial and terminal marginals, and its structure is amenable to both analytic characterization and efficient numerical solution. With limiting regimes precisely capturing both entropy-regularized and martingale optimal transport, SBB underpins theoretical advances and practical algorithms in transport, stochastic control, and modern generative modeling.

1. Problem Formulation and Analytical Structure

The dynamic SBB problem is specified as follows: for marginals $\mu, \nu \in \mathcal{P}_2(\mathbb{R}^d)$ and parameter $\beta > 0$, seek a continuous semimartingale

$X_t = X_0 + \int_0^t a_s\,ds + \int_0^t b_s\,dB_s$

where $B$ is a standard $d$-dimensional Brownian motion, $a_s \in \mathbb{R}^d$, $b_s \in \mathbb{R}^{d\times d}$ are progressively measurable and square-integrable, with initial and terminal constraints $X_0 \sim \mu$, $X_1 \sim \nu$. The SBB cost functional is

$$V_{\text{SB}}^\beta(\mu, \nu) = \inf_{X_\cdot} \mathbb{E} \left[ \int_0^1 \tfrac{1}{2}|a_t|^2 + \tfrac{\beta}{2}\|b_t - I_d\|_{\text{HS}}^2 \, dt \right],$$

penalizing both drift (kinetic energy) and deviation of the diffusion from the identity (volatility regularization) (Hasenbichler et al., 2 Apr 2026). The parameter $\beta > 0$0 controls the trade-off: as $\beta > 0$1, the problem becomes the Schrödinger bridge (with volatility $\beta > 0$2); as $\beta > 0$3 (after appropriate rescaling), the SBB converges to the martingale Benamou–Brenier (Bass) problem.

2. Weak Optimal Transport and Static Reformulation

A fundamental result is the reformulation of the SBB as a static weak optimal transport (WOT) problem, achieved via an explicit infimal convolution of the Schrödinger cost with a quadratic Wasserstein penalty. For any coupling $\beta > 0$4 with disintegration $\beta > 0$5, define the pointwise cost: $\beta > 0$6 This cost admits the representation

$\beta > 0$7

where $\beta > 0$8 denotes relative entropy, $\beta > 0$9 is the $X_t = X_0 + \int_0^t a_s\,ds + \int_0^t b_s\,dB_s$0 Gaussian, $X_t = X_0 + \int_0^t a_s\,ds + \int_0^t b_s\,dB_s$1 is the quadratic Wasserstein distance, and $X_t = X_0 + \int_0^t a_s\,ds + \int_0^t b_s\,dB_s$2 is the barycenter of $X_t = X_0 + \int_0^t a_s\,ds + \int_0^t b_s\,dB_s$3. This infimal convolution preserves the WOT structure, ensuring lower semicontinuity, coercivity, and stability of optimal couplings with respect to marginals (Hasenbichler et al., 2 Apr 2026).

The static SBB problem is then: $X_t = X_0 + \int_0^t a_s\,ds + \int_0^t b_s\,dB_s$4 Existence and uniqueness of optimal couplings hold under general conditions (Theorem 2.1 in (Hasenbichler et al., 2 Apr 2026)).

3. Primal–Dual Structure and Characterization of Optimizers

The static formulation directly yields strong duality, with primal and dual attainment. The SBB dual problem can be expressed as: $X_t = X_0 + \int_0^t a_s\,ds + \int_0^t b_s\,dB_s$5 with $X_t = X_0 + \int_0^t a_s\,ds + \int_0^t b_s\,dB_s$6, $X_t = X_0 + \int_0^t a_s\,ds + \int_0^t b_s\,dB_s$7 denoting infimal convolution, and

$X_t = X_0 + \int_0^t a_s\,ds + \int_0^t b_s\,dB_s$8

The unique optimizer $X_t = X_0 + \int_0^t a_s\,ds + \int_0^t b_s\,dB_s$9 determines both drift and volatility via a conditional entropic minimizer that tilts the reference measure and constructs the optimal semimartingale through associated dual potentials. Specifically, given $B$0, one constructs a potential $B$1, its associated map, and a Föllmer process $B$2, with the optimal SBB process given by

$B$3

$B$4 being the heat-kernel smoothed entropic potential (Hasenbichler et al., 2 Apr 2026).

4. Algorithmic Approaches and Numerical Implementation

The explicit WOT structure of SBB enables the design of scalable algorithms. The paper introduces a Sinkhorn-type alternating maximization algorithm for the dual formulation. Each iteration consists of

$B$5

followed by

$B$6

effectively solving an entropic–Wasserstein infimal convolution in each outer iteration. Each update increases the dual objective, and under mild integrability conditions, the sequence of dual potentials converges to the unique optimizer in the sense of epi-convergence (Hasenbichler et al., 2 Apr 2026).

5. Limiting Regimes: Schrödinger, Brenier–Strassen, and Bass

The SBB framework interpolates between classical transport regimes as the parameter $B$7 varies:

• As $B$8: $B$9, recovering the Schrödinger bridge cost and converging to the classical entropic optimal transport.
• As $d$0: $d$1, yielding the Brenier–Strassen weak quadratic transport problem. After rescaling by $d$2, the cost converges to the martingale Benamou–Brenier cost, restricting to martingale couplings.
• The SBB solution weakly converges to the stretched Brownian motion (the unique minimizer of the martingale Wasserstein cost) and, under irreducibility, to the Bass-martingale (Hasenbichler et al., 2 Apr 2026).

6. Continuity, Stability, and Weak Transport Structure

The static SBB cost is an infimal convolution of standard WOT-costs; this construction ensures the cost remains lower-semicontinuous, jointly convex in its variables, continuous on compact sets, and coercive. Importantly, the structure is robust under perturbations of the input marginals, with optimal transport plans varying continuously (Hasenbichler et al., 2 Apr 2026). This robustness enables efficient and stable numerical implementation and ensures reliability in practical generative and stochastic control applications.

7. Connections and Broader Significance

SBB synthesizes entropic and martingale optimal transport paradigms, providing a unified analytic and algorithmic foundation that bridges key problems in stochastic control, diffusion models, and weak transport. It yields a complete variational characterization, dual attainment, and practical algorithms; connects tightly with PDE approaches and entropy-regularized flows; and its limiting regimes clarify the relationships among canonical couplings in the theory of optimal transport. Applications include generative modeling, robust data-driven synthesis under structural constraints, and fine calibration of drift and volatility in financial and physics-based time series (Hasenbichler et al., 2 Apr 2026, Henry-Labordere et al., 29 Mar 2026, Alouadi et al., 25 Jan 2026, Alouadi et al., 8 Apr 2026).

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Table: Limiting Regimes of the SBB Cost

$d$3 Regime	Limiting Cost Structure	Limiting Problem
$d$4	$d$5	Schrödinger bridge (entropic OT)
$d$6	$d$7	Brenier–Strassen (weak quadratic OT)
$d$8	$d$9	Martingale Benamou–Brenier (Bass)

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The SBB formalism thus provides a rigorous and flexible tool in modern probabilistic transport, enabling structured interpolation between entropy-dominated and martingale-dominated regimes within a general weak optimal transport setting (Hasenbichler et al., 2 Apr 2026).