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We examine the hypothesis that every particle of mass m is subject to a Brownian motion with diffusion coefficient 2m and no friction. The influence of an external field is expressed by means of Newton's law F=ma, as in the Ornstein-Uhlenbeck theory of macroscopic Brownian motion with friction. The hypothesis leads in a natural way to the Schr\"odinger equation, but the physical interpretation is entirely classical. Particles have continuous trajectories and the wave function is not a complete description of the state. Despite this opposition to quantum mechanics, an examination of the measurement process suggests that, within a limited framework, the two theories are equivalent.
Edward Nelson (1966) studied this question.