Nelson derived the Schrodinger equation from a diffusion, so the same continuity equation that lets Bohm read a velocity off the wavefunction lets Nelson read a drift with noise against it, and the Born density solves both.
nelson_fokkerPlanckQuantum mechanics conserves probability, and the statement of that conservation is a continuity equation: the rate at which the probability of finding a particle in a region changes equals the flow of probability across the region's boundary. The equation is ordinary bookkeeping, and what's interesting is that it can be read in more than one way.
Divide the probability flow by the density and you get a velocity. Bohm's reading takes that velocity literally: there is one particle, it follows that velocity, and if its starting position is drawn from the quantum density then it stays distributed that way forever, which is what the continuity equation guarantees. Nelson's reading splits the same flow differently, into a drift plus a term that looks exactly like diffusion, and then asks for a random process with that drift and that amount of noise. The Born density solves the resulting diffusion equation, so a particle jiggling according to Nelson's rules is distributed as quantum mechanics says at every moment.
Two very different pictures, one identity. That's worth knowing before arguing about which picture is right, because the shared mathematics is not in dispute and the disagreement is about what the pieces mean.
Readers who know Bohmian mechanics ask how a programme with one trajectory and no ontological wavefunction differs from Bohm, and the corpus answers by proving the shared mathematics rather than asserting it. The continuity equation is proved from the Schrodinger equation at a point, the polar form gives the current as the density times the phase gradient, which is Bohm's equivariance, the same statement is given in Lagrangian form along a trajectory, and Nelson's osmotic drift turns the identity into the Fokker-Planck equation of his diffusion.
The differences are then stated where they belong, as positioning rather than as theorems. Bohm's configuration is guided by a wavefunction that is a second ontological ingredient with its own law; the programme's trajectory lives on the constraint surface and the wavefunction labels an epistemic region. Bohm's dynamics reads a velocity off the phase; the programme's isolated dynamics is the Hamiltonian flow of the sector's own symplectic structure. Bohm samples the initial position from the quantum density by hypothesis and equivariance keeps it there; the programme derives the weights as typicality volumes of an invariant measure. The diffusion process itself is not claimed, as stochastic differential equations aren't available at the pin, and relaxation of a non-equilibrium density is a numbered open row in both settings.
In one dimension with the mass and Planck constant set to one, for psi satisfying the Schrodinger equation at the point in question with real potential: hasDerivAt_probCurrent gives the current's divergence as Im(conj psi times the second derivative), hasDerivAt_probDensity_time gives the density's time derivative as twice Re(conj psi times the time derivative), and continuity_equation is their cancellation. probCurrent_polar gives J = R^2 times the derivative of S in the polar form R e^(i S), so continuity_polar is the equivariance statement with velocity the phase gradient, and hasDerivAt_density_along_trajectory is its Lagrangian form.
nelsonFlux is rho v + (1/2) times the derivative of rho, nelsonFlux_eq_mul_drift writes it as rho times the osmotic drift where the density doesn't vanish, and nelson_fokkerPlanck is the rearrangement showing the continuity equation is exactly the Fokker-Planck equation with diffusion coefficient one half. Nelson's theorem that a diffusion with that drift has the Born density as its law at every time needs stochastic differential equations and is not claimed, carried as a machine-checked absence sentinel; global existence of Bohmian trajectories, singular at the nodes, is not attempted; one dimension only. The positioning is specs/bohm-nelson-scoping.md.
Edward Nelson published the derivation in 1966 at Princeton, showing that a particle in a diffusion with a particular drift obeys the Schrodinger equation, and later came to doubt the programme himself on the grounds that the stochastic picture fails to reproduce quantum correlations across separated systems. His book Quantum Fluctuations of 1985 is the developed version.
Erwin Madelung had written the hydrodynamic form of the Schrodinger equation as early as 1927, Louis de Broglie proposed the pilot-wave reading in the same period, and David Bohm revived and completed it in 1952.
CsdLean4/Mathlib/Analysis/Calculus/SchrodingerCurrent.leanSource links are pinned to a commit, so they do not drift. The anchors above are checked mechanically against the Lean tree on every build. The mathematics is not, and cannot be: that is a human responsibility and it rests with the author.
Part of Constraint-Surface Dynamics · Formalised in csd-lean4.
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