Reflected diffusion, no-flux continuity equations and confined Lagrangian flows in bounded domains

Technical Summary: Reflected Diffusions, No-Flux Continuity Equations, and Confined Lagrangian Flows in Bounded Domains

Problem Statement
The paper investigates the mathematical relationship between the Eulerian description (density evolution) and the Lagrangian description (deterministic particle trajectories) of probability flows in bounded domains. Specifically, it addresses whether a density–flux pair $(p, J)$ solving a no-flux continuity equation admits a Regular Lagrangian Flow (RLF) that remains confined within the closed domain $\bar{\Omega}$ and transports the initial measure to the prescribed marginals.

This problem is motivated by constrained generative modeling using reflected diffusion processes. In such models, a forward process is a reflected stochastic differential equation (SDE) confined to a domain $\Omega$. Sampling is typically performed by reversing the density flow. While it is known that the marginal densities of the reflected SDE satisfy a Fokker–Planck equation with no-flux boundary conditions, it is often assumed that these same marginals can be generated by a deterministic Probability Flow ODE (PF-ODE) driven by the velocity field $v = f - \frac{1}{2}\sigma^2 \nabla \log p$. A critical gap exists in the literature: the PF-ODE contains no explicit reflection term, yet it must generate trajectories that never leave $\bar{\Omega}$. The paper seeks to establish precise conditions under which this "mimicking" holds and to identify when it fails.

Methodology
The analysis relies on the Ambrosio–DiPerna–Lions theory for continuity equations with low-regularity vector fields. The core methodological tool is the zero-extension technique: extending the velocity field $v$ by zero outside the domain $\Omega$ to a field $v_0$ on $\mathbb{R}^d$.

1. Boundary Sheet Identity: The paper utilizes a distributional identity for the divergence of the zero extension of a BV field. The divergence of the extended field $v_0$ contains a singular part supported on the boundary $\partial\Omega$ proportional to the normal trace of $v$:
   $$D \cdot v_0 = (\nabla \cdot v) \mathcal{L}^d \llcorner \Omega - (\text{tr } v \cdot \nu) \mathcal{H}^{d-1} \llcorner \partial\Omega$$
   For the Ambrosio–DiPerna–Lions theory to apply (ensuring existence and uniqueness of an RLF), the singular boundary part must vanish. This requires the tangency condition: $\text{tr } v \cdot \nu = 0$ almost everywhere on the boundary.

2. Weighted Energy Methods: To handle singular drifts (specifically "entrance-type" singularities where the drift blows up at the boundary), the paper develops a weighted energy uniqueness result for Fokker–Planck equations. This involves testing the equation for the ratio $q/p$ against itself in a weighted space $L^2(p dx)$, leading to an exact cancellation of transport terms and avoiding the need for the drift to be bounded or have a bounded divergence.

3. Rigidity and Counterexamples: The paper constructs explicit smooth counterexamples to test the sharpness of the assumptions, analyzing the interplay between boundary currents, compressibility, and the existence of integral curves.

Key Contributions and Results

1. Sufficient Conditions for Confined RLF (Theorem A)
The paper establishes that a no-flux weak solution $(p, J)$ admits a unique RLF confined to $\bar{\Omega}$ if the following conditions hold:

• Regularity: The velocity $v = J/p$ is in $L^1([0, T]; BV_{loc}(\Omega))$ and has bounded variation in a collar near the boundary.
• Divergence Control: The negative part of the divergence $[\nabla \cdot v]^-$ is in $L^1_t L^\infty_x$.
• Tangency: The normal trace of the velocity vanishes on the boundary ($\text{tr } v \cdot \nu = 0$).

Under these conditions, the zero extension of $v$ satisfies the hypotheses of the Ambrosio–DiPerna–Lions theorem. The proof shows that trajectories of the extended field cannot exit the domain (as they would be frozen outside, contradicting continuity), and the flow transports the initial density to the marginal densities.

2. The Boundary Current Mechanism (Theorem B)
The paper constructs an explicit smooth density–flux pair on a periodic strip that solves the no-flux continuity equation but fails to admit a Regular Lagrangian Flow.

• Mechanism: The example features a "boundary current": a tangential mass current of non-vanishing linear density along a wall where the volume density vanishes linearly.
• Failure Mode: While the density evolution is unique (in a weighted energy class) and integral curves exist and are confined, the flow fails the compressibility bound required for an RLF. The Jacobian of the flow blows up arbitrarily close to the initial time, meaning no single compressibility constant can bound the push-forward of the Lebesgue measure.
• Significance: This demonstrates that the boundary assumptions (tangency, BV regularity near the boundary) are structurally entangled and cannot be jointly relaxed to admit such boundary currents.

3. Rigidity Results
The paper proves that in the class of smooth, non-degenerate boundary vacuums:

• Tangency is Automatic for Bounded Velocities: If the velocity is bounded near a $C^1$ wall where density vanishes linearly, the no-flux condition forces the velocity to be tangential. Thus, failure of tangency implies the velocity must be unbounded (specifically, blowing up like $1/\text{dist}$).
• One-Dimensional Rigidity: In 1D, invariance (trajectories staying in the domain) is guaranteed by mass conservation and the no-flux condition; failure can only occur via non-existence of curves from vacuum initial data, not via boundary exit.

4. Application to Reflected Diffusions (Corollaries 5.1, 5.4)
The paper applies Theorem A to reflected diffusion models:

• Early Stopping: For reflected diffusions with parabolic Hölder regular coefficients, the density becomes $C^{2,\alpha}$ and strictly positive for any $t > 0$. Consequently, the tangency condition holds pointwise, and the probability-flow ODE generates a valid confined RLF on any interval $[\delta, T]$ with $\delta > 0$.
• Data Endpoint Failure: At $t=0$, if the initial data has a vacuum (e.g., compact support), the velocity field may fail to be integrable near the boundary, preventing the existence of an RLF starting from time zero. This mirrors whole-space counterexamples but is driven by the interaction with the reflecting boundary.

Significance and Claims
The paper provides a rigorous mathematical justification for using reflection-free ODEs to sample from reflected diffusion models, provided that:

1. Early stopping is employed to avoid singularities at $t=0$.
2. The combined drift–score field satisfies specific regularity and boundary tangency conditions.

The work clarifies that the "mimicking" of reflected diffusions by deterministic flows is not automatic; it relies on the vanishing of the singular boundary term in the divergence of the zero-extended velocity. The paper also highlights a fundamental gap between Eulerian well-posedness (uniqueness of density evolution, which holds for bounded measurable drifts) and Lagrangian well-posedness (existence of a stable deterministic flow, which requires stronger regularity and boundary conditions). The results indicate that deterministic samplers may fail if the boundary current mechanism is activated or if the initial data is too singular, even if the marginal densities are uniquely determined.