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

Motivated by marginal distribution flows of reflected diffusions in bounded domains, we investigate when a density/flux pair solving a no-flux continuity equation admits a regular Lagrangian flow that remains in the closed domain and generates the prescribed density flow. We give sufficient conditions in terms of interior bounded-variation regularity, bounded-variation control on a boundary collar, a one-sided bound on an absolutely continuous divergence, and vanishing normal trace of the velocity. The proof uses the fact that tangency removes the singular boundary contribution to the divergence of the zero extension, thereby making the extended velocity admissible for the Ambrosio-DiPerna-Lions theory. We show that these boundary assumptions cannot be jointly relaxed so as to admit a boundary current mechanism. We construct an explicit smooth density/flux pair carrying a boundary current. Its density evolution is unique in a weighted class and its characteristics are unique, confined and transport the marginals, yet it admits no regular Lagrangian flow because the compressibility bound fails arbitrarily close to the initial time. We also establish two uniqueness results for no-flux Fokker-Planck equations: a duality result for bounded measurable drifts and a weighted energy result for entrance-type drifts singular at the boundary. Our results provide a rigorous mathematical justification for using the ODE-based sampling of reflected diffusion models under minimal regularity assumptions on the coefficients, and also indicate when such ODE-based samplers may fail.