A PDE approach to Benamou--Brenier formula for the Schrödinger problem

1. Problem Statement and Motivation

The Schrödinger Problem:
The paper addresses the Schrödinger problem, a variational problem originating from statistical mechanics. Given two probability measures $\mu_0, \mu_1$ on $\mathbb{R}^n$ and a parameter $\varepsilon > 0$ (representing temperature or noise), the goal is to find the most likely evolution of a Brownian particle cloud connecting these two states. Mathematically, this is formulated as minimizing the relative entropy with respect to a reference measure $R_\varepsilon$ (the joint law of a reversible Wiener process):
$$\inf \{ H(\gamma | R_\varepsilon) : \gamma \in \Gamma(\mu_0, \mu_1) \}$$
where $\Gamma(\mu_0, \mu_1)$ is the set of transport plans with marginals $\mu_0$ and $\mu_1$.

The Gap in Existing Theory:
The Schrödinger problem is known to be an entropic regularization of the quadratic Optimal Transport (OT) problem. In OT, the Benamou–Brenier formula provides a dynamic reformulation, equating the static transport cost to the minimum kinetic energy of a fluid flow satisfying the continuity equation.
For the Schrödinger problem, a similar dynamic formulation exists involving the Fokker–Planck equation:
$$\partial_t \eta_t - \frac{\varepsilon}{2}\Delta \eta_t + \text{div}(v_t \eta_t) = 0$$
However, previous rigorous proofs of this dynamic formula (specifically by Gigli and Tamanini, [GT20]) relied on compactness assumptions: the marginals $\mu_0$ and $\mu_1$ were required to have bounded support.
The Issue: Many practical applications (e.g., Gaussian models, mixture-of-Gaussians, Bayesian inference) involve measures with unbounded support. In these cases, the Schrödinger potentials ($f_\varepsilon, g_\varepsilon$) and their logarithms ($\varphi_t^\varepsilon, \psi_t^\varepsilon$) may exhibit exponential growth at infinity, making the standard integration-by-parts arguments and differentiation under the integral sign invalid without careful control.

2. Methodology

The authors propose a PDE-based approach to bridge the gap between abstract theory and non-compact applications. Their strategy involves three main pillars:

1. Relaxing Marginal Assumptions: Instead of compact support, they assume the marginals are of the form $\mu_i = e^{-V_i} \mathcal{L}^n$, where $V_i \in C^2(\mathbb{R}^n)$ satisfy uniform two-sided Hessian bounds:
   $$C^{-1}|\xi|^2 \leq \xi \cdot D^2 V_i(x) \xi \leq C|\xi|^2$$
   This class includes Gaussian measures and strongly log-concave distributions (sub-Gaussian tails).

2. Regularity and Growth Estimates:

   • They leverage recent results (from [Chi+24] and [Con24]) to establish that the Schrödinger potentials $\psi_t^\varepsilon = \varepsilon \log g_t^\varepsilon$ inherit Hessian bounds from the marginals.
   • They prove that $\psi_t^\varepsilon$ grows at most quadratically, and its gradient $\nabla \psi_t^\varepsilon$ grows at most linearly.
   • Crucially, they show that the entropic interpolation $\varrho_t^\varepsilon$ (the density of the path) possesses strong Gaussian moments that decay fast enough to compensate for the polynomial/exponential growth of the potentials.

3. Rigorous PDE Analysis:

   • Differentiation under the Integral: They use the derived moment bounds to justify differentiating $\int \psi_t^\varepsilon \varrho_t^\varepsilon dx$ with respect to time.
   • Integration by Parts (Gauss–Green): They construct specific cut-off functions and use Parabolic Kato's inequality to prove that boundary terms vanish at infinity, allowing the application of Gauss–Green formulas for unbounded domains.
   • Uniqueness via Convexity: They establish the uniqueness of the velocity field in the dynamic formulation using the strict convexity of the kinetic energy functional and the method of characteristics.

3. Key Contributions

• Extension to Sub-Gaussian Measures: The paper extends the validity of the Benamou–Brenier formula for the Schrödinger problem from compactly supported measures to a broad class of measures with sub-Gaussian tails (including Gaussians and mixtures).
• Almost Self-Contained PDE Proof: Unlike previous proofs that relied on probabilistic arguments or compactness, this work provides a rigorous, almost self-contained proof using only PDE tools (heat kernel estimates, Hessian bounds, and moment controls).
• Explicit Growth Control: The authors provide explicit estimates showing that even if the potentials grow at infinity, the decay of the entropic interpolation density ensures the necessary integrability for the dynamic formulation to hold.
• Uniqueness Result: They prove that the minimizer of the dynamic kinetic action is unique under the Fokker–Planck constraint.

4. Main Results

Theorem 1.1 (Simplified Benamou–Brenier Formula):
Under the assumption that $\mu_0, \mu_1$ are sub-Gaussian measures (defined by $C^2$ potentials with bounded Hessians), for every $\varepsilon > 0$:
$$\varepsilon \min_{\gamma \in \Gamma(\mu_0, \mu_1)} H(\gamma | R_\varepsilon) = \varepsilon H(\mu_0 | \mathcal{L}^n) + \min \left\{ \int_0^1 \int_{\mathbb{R}^n} \frac{|v_t|^2}{2} \eta_t \, dx dt \right\}$$
where the minimum is taken over pairs $(\eta_t, v_t)$ satisfying the Fokker–Planck equation in the distributional sense:
$$\partial_t \eta_t - \frac{\varepsilon}{2}\Delta \eta_t + \text{div}(v_t \eta_t) = 0, \quad \eta_0 = \mu_0, \quad \eta_1 = \mu_1$$
The unique minimizer for the kinetic energy is given by the entropic interpolation path $(\mu_t^\varepsilon, \nabla \psi_t^\varepsilon)$.

Corollary (Dual Formulation):
The paper also derives a dual formulation for the Schrödinger problem involving a Hamilton-Jacobi-Bellman type inequality:
$$\max \left\{ \int f_1 d\mu_1 - \int f_0 d\mu_0 : \partial_t f + \frac{\varepsilon}{2}\Delta f + \frac{1}{2}|\nabla f|^2 \leq 0 \right\}$$
where the maximizer is the Schrödinger potential $\psi_t^\varepsilon$.

5. Significance and Impact

• Bridging Theory and Practice: The result justifies the use of dynamic formulations (fluid-dynamical interpretations) in settings where data is naturally unbounded, such as Gaussian mixture models, diffusion models in machine learning, and stochastic control.
• Algorithmic Implications: The Schrödinger problem is the theoretical basis for the Sinkhorn algorithm, a cornerstone of modern optimal transport and machine learning. Validating the dynamic formulation for unbounded supports strengthens the theoretical foundation for applying these algorithms to real-world, non-compact datasets.
• Analytical Rigor: By providing a PDE-centric proof that handles unbounded domains via moment estimates, the paper offers a robust framework for analyzing entropic interpolations beyond the restrictive compact setting, opening avenues for further research in non-compact optimal transport and stochastic analysis.

In summary, this paper resolves a critical technical limitation in the theory of the Schrödinger problem, proving that the elegant dynamic connection between entropy minimization and kinetic energy holds for the wide class of sub-Gaussian distributions essential to modern applications.