Abstract
We take a new look at the relation between the optimal transport problem and the Schrödinger bridge problem from the stochastic control perspective. We show that the connections are richer and deeper than those previously described in existing literature. In particular: a) We give an elementary derivation of the Benamou-Brenier fluid dynamics version of the optimal transport problem; b) We provide a new fluid dynamics version of the Schrödinger bridge problem; c) We observe that the latter provides an important connection with optimal transport without zero noise limits; d) We propose and solve a fluid dynamic version of optimal transport with prior; e) We can then view optimal transport with prior as the zero noise limit of Schrödinger bridges when the prior is any Markovian evolution. In particular, we work out the Gaussian case.
Summary
This paper provides a unified stochastic control perspective on the relationship between optimal mass transport (OMT) and the Schrödinger bridge problem. The central insight is that both problems can be formulated as fluid dynamic control problems with the same structure — minimize the kinetic energy of a velocity field subject to a continuity equation and boundary marginals — but the Schrödinger bridge functional includes an additional Fisher information penalty term (1/8)∫||∇ln ρ||²ρ dx. This reveals that the two problems are related without needing zero-noise limits: the SB problem is OMT plus a Fisher information regularization.
The paper introduces the concept of optimal transport with prior (Section VII), where one minimizes the deviation of a velocity field from a given prior velocity v(x,t), subject to continuity and marginal constraints. This generalizes both OMT (v≡0) and provides the zero-noise limit of Schrödinger bridges when the prior is any Markovian evolution. In the Gaussian case (Section VIII), everything reduces to matrix Riccati equations, and the convergence of Schrödinger bridges to OMT with prior as ε→0 is made explicit.
The stochastic control formulation (Section IV) is particularly relevant to bridge-based generative modelling: the optimal drift of the Schrödinger bridge is b_+^{Q*} = b_+^P + ∇ln φ, where φ is space-time harmonic under the prior. This is precisely the score correction that appears in score-based generative models, and connects directly to the generalized Tweedie’s formula used in Connecting Brownian and Poisson Random Bridges with Rectified Flows — the bridge drift correction ∇ln φ plays the role of the Tweedie score correction σ²∇log f(z) that adjusts the prior drift toward the posterior.
Key Contributions
- Elementary derivation of the Benamou-Brenier formula for OMT as a stochastic control problem
- New fluid dynamic formulation of the Schrödinger bridge problem (eq 54-56)
- Shows SB and OMT differ by a Fisher information functional — no zero-noise limit needed
- Introduces optimal transport with prior as a new variational problem (eq 59)
- Shows OT with prior is the zero-noise limit of SB with non-trivial Markovian prior
- Time-symmetric formulation via Nelson’s current and osmotic drifts (Section V)
- Explicit Gaussian theory via matrix Riccati equations (Section VIII)
Methodology
The approach uses Girsanov’s theorem to express the relative entropy H(Q,P) as the expected integrated kinetic energy of the drift difference (eq 20a-b). The SB problem min H(Q,P) then becomes a stochastic control problem: find the optimal drift perturbation u(x,t) to the prior drift b_+^P that minimizes E_Q[∫½||u||²dt], subject to the controlled SDE dx = [b_+^P + u]dt + dw and marginal constraints. The optimal control u* = ∇ln φ where φ satisfies the space-time harmonic equation ∂φ/∂t + b_+^P·∇φ + ½Δφ = 0 (eq 33). The fluid dynamic version follows by converting to the continuity equation formulation via v = (b_+ + b_-)/2 (current velocity) and u = (b_+ - b_-)/2 = ½∇ln ρ (osmotic velocity).
Key Findings
- The Benamou-Brenier formulation: inf_{(ρ,v)} ∫∫½||v||²ρ dtdx subject to ∂ρ/∂t + ∇·(vρ) = 0, ρ(0)=ρ₀, ρ(1)=ρ₁
- The SB fluid dynamic analogue: inf_{(ρ,v)} ∫∫[½||v||² + ⅛||∇ln ρ||²]ρ dtdx subject to the same constraints
- The two differ exactly by the Fisher information term ⅛∫||∇ln ρ||²ρ dx
- Optimal transport with prior: inf_{(ρ̃,ṽ)} ∫∫½||ṽ - v||²ρ̃ dtdx, minimizing deviation from prior velocity v
- In the Gaussian case, ε→0 convergence of SB to OT with prior is governed by matrix Riccati equations
- The optimal SB drift decomposes as prior drift + ∇ln φ (forward) or prior drift - ∇ln ψ (backward)
Important References
- Benamou-Brenier formula — Fluid dynamic formulation of optimal transport that this paper parallels for Schrödinger bridges
- Optimal Transport — Old and New — Villani’s comprehensive treatment of optimal transport theory
- Stochastic Control and Schrödinger Bridges — Léonard’s survey connecting SB to OT via large deviations