A Regime-Switching Approach to the Unbalanced Schrödinger Bridge Problem (2512.12971v1)

Abstract: The unbalanced Schrödinger bridge problem (uSBP) seeks to interpolate between a probability measure $ρ_0$ and a sub-probability measure $ρ_T$ while minimizing KL divergence to a reference measure $\mathbf{R}$ on a path space. In this work, we investigate the case where $\mathbf{R}$ is the path measure of a diffusion process with killing, which we interpret as a regime-switching diffusion. In addition to matching the initial and terminal distributions of trajectories that survive up to time $T$, we consider a general constraint $ψ(t,x)$ on the distribution of killing times and/or killing locations. We investigate the uSBPs corresponding to four choices of $ψ$ in detail which reflect different levels of information available to an observer. We also provide a rigorous analysis of the connections and the comparisons among the outcomes of these four cases. Our results are novel in the field of uSBP. The regime-switching approach proposed in this work provides a unified framework for tackling different uSBP scenarios, which not only reconciles but also extends the existing literature on uSBP.

• The paper establishes a regime-switching formulation for the unbalanced Schrödinger Bridge Problem, unifying various observational settings and extending existing models.
• It derives explicit stochastic differential equations and harmonic solutions that ensure uniqueness and Markov properties under state-dependent killing.
• The framework offers robust computational strategies, including a Fortet-Sinkhorn algorithm, with significant implications for modeling processes with mass dissipation.

Regime-Switching Formulation of the Unbalanced Schrödinger Bridge for Diffusions

Introduction and Context

The unbalanced Schrödinger Bridge Problem (uSBP) seeks an entropic interpolation between an initial probability measure $\rho_0$ and a sub-probability measure $\rho_T$ by minimizing the Kullback-Leibler (KL) divergence with respect to a reference path measure $R$. The reference dynamics are given by a diffusion process with state-dependent killing (i.e., mass loss). This paper establishes a comprehensive theoretical framework for the diffusion uSBP by recasting it as an SBP with regime switching, thus leveraging the analytic machinery of regime-switching diffusions (Zlotchevski et al., 8 Nov 2025). This approach not only recovers and unifies existing results in the literature but also extends the problem class and provides general tools for further analysis and computation.

Mathematical Framework

Regime-Switching Diffusion with Killing

The core mathematical structure consists of:

• State Space: $\mathbb{R}^d \times \{\text{active}, \text{dead}\}$, with evolution modeled on cà dlà g path space, including both living (active) and killed (dead) particle regimes.
• Dynamics: Particles evolve as solutions to SDEs until a random killing time $K$, determined by a rate $V(t,x)$, at which point the process switches to the dead regime and possibly jumps to a new spatial coordinate specified by $\psi(K, X_{K-})$.
• Reference Measure: $R$ is the path law of such a regime-switching diffusion, which admits positive transition densities and is Markov.

Schrödinger System Formulation

The uSBP seeks a path measure $\widehat{P}$ minimizing $\mathrm{KL}(P\,|\,R)$ under marginal constraints at $t=0$ ($\rho_0$) and $t=T$ ($\rho_T$), now possibly sub-probability at the terminal time. The optimal path law can always be represented as

$$\widehat{P} = f(X_0, \Lambda_0) \, g(X_T, \Lambda_T) \, R$$

where $(f, g)$ is the solution of an abstract Schrödinger system coupling forward evolution with backward conditioning. This solution defines explicit SDE and killing rate dynamics under $\widehat{P}$.

The regime-switching formalism allows explicit representation of the generator, forward/backward equations, and transition probabilities under $\widehat{P}$ in terms of those for $R$, modulated by harmonic functions $\varphi, \widehat{\varphi}$ obtained from the Schrödinger system.

Regime-Switching Approach: Four Observational Models

The paper provides a taxonomy of meaningful observational scenarios in the uSBP classification, each corresponding to different choices of $\psi$ (the spatial jump at killing) and the marginal constraints imposed at $t = T$:

1. Full Spatio-temporal Killing Distribution: Observer knows joint distribution of killing time and location.
2. Temporal Killing Distribution: Observer knows only the distribution of killing times.
3. Spatial Killing Distribution: Observer knows only the distribution of killing locations.
4. Total Killed Mass Only: Observer knows only the total killed mass (original, unbalanced SBP).

Each case is formalized as a distinct Schrödinger bridge problem, with a precise characterization of the associated terminal constraints, state augmentation, and admissible measures.

Analytical and Structural Results

Existence, Uniqueness, and Structure

The authors give regularity and existence conditions ensuring that the uSBP solution exists, is unique, and maintains a Markov property (i.e., the solution is again a regime-switching diffusion). The generator, transition densities, and Kolmogorov equations for $\widehat{P}$ are explicitly described, including the effect of the harmonic functions $(\varphi, \widehat{\varphi})$ arising from the Schrödinger system.

Stochastic Control Interpretation

The uSBP admits a variational interpretation as an optimal control problem in which the drift and killing rate of the dynamics are controlled, with the cost functional being the cumulative relative entropy between controlled and nominal (reference) dynamics. The optimal controls are expressed in closed form via the Schrödinger system, echoing classical results but now generalized to processes with irreversible mass loss (killing).

Case Analysis and Algorithmic Aspects

For each of the four observational scenarios, the paper details the structure of the marginal distributions in the dead regime and derives explicit forms for the backward harmonic functions. These explicit formulas yield both theoretical insights (e.g., support and regularity of the killed mass marginals) and practical computation strategies using the Fortet-Sinkhorn algorithm.

Comparison Principles and Hierarchy of Constraints

A major analytical result is a hierarchy/comparison theorem relating the KL optimal values and solution forms across the four regimes and their inter-relations. Central statements include:

• Stronger (more informative) constraints (e.g., knowing the full spatio-temporal distribution) always yield a higher minimal KL divergence compared with weaker constraints.
• Equality in KL costs across models occurs if and only if specific marginalizations of the more informative target distribution coincide with the less informative constraints, leading to solutions with degenerate ("constant") Schrödinger factors.

The precise conditions for such equalities are computed, showing that for Dirac initial data, the marginal equivalence is automatic.

Connections to Previous Works and Extensions

The regime-switching framework allows seamless connection to prior models:

• It recovers [Chen-Georgiou-Pavon 2022]'s treatment of the original unbalanced Schrödinger bridge as the special case with only total killed mass observed.
• The setting also encompasses and clarifies the marginal-only constraint models of Eldesoukey et al., showing that in these cases, certain Schrödinger system solutions collapse to constants.
• Extensions to incomplete/foggy observations, mixed-marginal observational models, and processes with possible revival (reverse switching) are shown to be natural in the regime-switching language.

Theoretical and Practical Implications

The regime-switching approach yields a general platform for analyzing and computing unbalanced Schrödinger bridges for killed diffusions, subsuming numerous previous SBP-like formulations. By deriving explicit forms for generators and harmonic systems, the framework enables:

• Qualitative analysis of the effect of different levels of observational constraints on entropic flows in open systems with mass dissipation.
• Robust computational methods (via iterative Schrödinger solvers) for both forward simulation and inverse inference problems (e.g., reconstructing most likely history of mass loss in observed dynamical systems).

On the theoretical side, these results clarify the structural connection between unbalanced optimal transport, entropy minimization, and controlled PDMPs (piecewise deterministic Markov processes). On the practical side, the approach opens avenues for flexible modeling in domains where partial or noisy observations of mass extinction or transformation events are central (statistical physics, cell biology, population dynamics, etc.), as well as for scalable unbalanced SB solvers for high-dimensional inference in data science.

Conclusion

This work advances the theory of the unbalanced Schrödinger Bridge Problem by embedding it within the theory of regime-switching diffusions, showing that a unified analytical and algorithmic structure emerges across a landscape of marginal observation scenarios. The regime-switching formalism offers a principled route for both theoretical analysis and computational implementation, and supports further extensions to non-standard observational settings and dynamics with richer event structures. The implications are significant for both the mathematical theory of entropy minimization in open systems and for computational approaches to generative modeling with mass dissipation (2512.12971).