This is what the field of numerical analysis exists for. These details definitely have been treated, but this was done mainly early in the field's history; for example, by people like Wilkinson and Kahan...
So I guess one might want to do a similar exercise to deriving numerical dispersion for example in order to see just how discretizing the diffusion process affects it and the relation to optimal control theory.
[1]: https://en.wikipedia.org/wiki/Numerical_dispersion
[2]: https://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%...