I don't mean to mean but: what is surprising about any of this ?
Joseph Fourier's solution to the heat-equation (linear diffusion) was in fact the origin of the FT. The high-freq coefficients decay (as -t^2 IIRC) in there; the reverse is also known to be "unstable" (numerically, and is singular from the equillibrium).
More over, the reformulation doesn't immediately reveal some computational speedup, or a better alternative formulation (which is usually a measure of how valuable it is epistemically).
(Edit: note that Heat-equation is more akin to the Fokker-Planck eqn, not actual Diffusion as an SDE as is used in Diffusion models).