Nelsonian mechanics revisited

In de Broglie and Bohm's pilot-wave theory, as is well known, it is possible to consider alternative particle dynamics while still preserving the \psi\(2) distribution. I present the analogous result for Nelson's stochastic theory, thus characterising the most general diffusion processes that preserve the quantum equilibrium distribution, and discuss the analogy with the construction of the dynamics for Bell's beable theories. I briefly comment on the problem of convergence to \psi\(2) and on possible experimental constraints on the alternative dynamics.