Continuity equations and the Benamou–Brenier formula

Let the endpoints belong to P₂ of real d-space. Use admissible curves ρ in C([0, 1]; P₂) for the W₂ metric and Borel velocity fields vₜ with finite integral of |vₜ|² against ρₜ dt, satisfying the continuity equation in distributions with the prescribed endpoints. W₂ continuity includes continuous bounded-test averages and matching second-moment continuity. The finite-action continuity equation then gives an absolutely continuous metric curve, with square-integrable metric speed bounded by the velocity norm.

The equation says that probability changes only through transport flux, not creation or destruction. In weak form, differentiation of an observable integrates its spatial gradient against velocity and mass, with the corresponding endpoint terms for time-dependent tests. This formulation includes atoms following differentiable paths. Writing a classical density derivative would wrongly exclude them. Velocities are determined only where the measure places mass; their values on empty regions cannot change the action.

The dynamic theorem identifies W₂ squared with the infimum of kinetic action over these admissible pairs. Our static ground cost is |x − y|², so no factor one-half appears. A time interval of length L would change the minimum action to W₂ squared divided by L; moving the same distance more quickly costs more. The unit interval is therefore an essential part of the displayed formula.

An optimal static coupling supplies an upper-bound construction by straight particle paths. Jensen’s inequality relates a path’s squared net displacement to its integrated squared speed, suggesting the reverse bound. Turning an arbitrary admissible measure curve into such particle paths requires a superposition theorem or the full dynamic proof; this paragraph is motivation, not an elementary proof of that representation. Under the stated Euclidean P₂ framework the theorem provides equality and the constant-speed displacement curve is a minimizer.

When a path has a density r and flux m = rv, the integrand becomes |m|²/r. Its lower-semicontinuous extension is zero at (r, m) = (0, 0) and infinite when r = 0 but m is nonzero. This perspective function is jointly convex, while the conservation law is linear in density and flux. Those facts motivate convex numerical formulations; choosing velocity and density as independent variables obscures this structure.

The density notation is a useful special case, not permission to ignore singular endpoints or weak boundary conditions. A mesh-based implementation needs a specific discrete conservation law and a convention at zero density. Numerical diffusion may alter the represented path even when the endpoint histograms look right. Before interpreting a computed action as W₂ squared, verify conservation, endpoint constraints, and discretization error rather than only a visually smooth animation.