Kurtz Theorems: two limits, timescale and system size, often taken in the wrong order in practice.
Einstein-Smoluchowski debate regarding Langevin correspondence to Ito vs Statonvich representations resolved: [theorem]. (Statonvich is the more natural interpretation of the way in which we measure data to compare to Langevin. Of course Ito and Stratonvich can be mapped back and forth anyway).
Of course this difference doesn’t exist, and nothing goes wrong / no stochastic surprises if dynamics are linear.
Necessary limits of applicability not often checked in SDE famework.
Deterministic limit often works better than these limits would imply, as the same deterministic equation arises from multiple cases
Hamiltonian-Jacobi derivation sheds some light on why deterministic result is more general (requires fewer assumptions/weaker limits than SDE approach)
Intrinsic + extrinsic noise is hard. Environmental noise poorly defined, hard to make physical. Can be incorporated into the master equation directly in a more meaningful way, still very hard problem. Still, environmental noise often not needed to capture the behavior once we go back to original master equation.