Incompatibility of the Schrödinger equation with Langevin and Fokker-Planck equations

Quantum mechanics posits that the wave function of a one-particle system evolves with time according to the Schrödinger equation, and furthermore has a square modulus that serves as a probability density function for the position of the particle. It is natural to wonder if this stochastic characterization of the particle's position can be framed as a univariate continuous Markov process, sometimes also called a classical diffusion process, whose temporal evolution is governed by the classically transparent equations of Langevin and Fokker-Planck. It is shown here that this cannot generally be done in a consistent way, despite recent suggestions to the contrary.