Stochastic mechanics & classical mechanics with finite differences

Stochastic mechanics develops a formulation of quantum physics by assuming that a classical particle is also subjected to Brownian-like motion. Because this presumes that a particle's path is non-differentiable; making this transformation often involves implicitly replacing time-derivatives in classical mechanics by finite differences. rn this paper, we will develop a stochastic mechanics by explicitly replacing time-derivatives in classical mechanics by finite differences, deriving the resulting dynamics and then introducing stochasticity. Like Nelson's original version of stochastic mechanics, our stochastic mechanics is also consistent with the Schrodinger Equation.