Criteria are proposed that will allow one to decide when a physical theory is a «stochastic mechanics». The standard formulation of stochastic quantization at zero temperature is shown to not satisfy these criteria. By analysing the incompatibilities with Fokker-Planck dynamics, stochastic quantization is modified to give a true stochastic mechanics. As a direct consequence, Davidson’s freedom in the diffusion constant disappears. Positive-temperature versions of stochastic quantization are also shown not to be a stochastic mechanics either. To do this, the Guerra-Loffredo version must be generalized from the case of the oscillator to arbitrary potentials with one minimum. A positive-temperature version of stochastic mechanics for the equilibrium situation is developed. Equilibrium fluctuations around the classical oscillator trajectory are shown to include Guerra and Loffredo’s coherent states and Skagerstam’s result on dissipative diffusion. General coherent states and connections with Euclidean quantum mechanics are also discussed.