The diffusion equation has a dirty secret: it doesn't survive a change of reference frame.
This has been known for decades. Boost a diffusing system — view it from a moving observer's perspective, as relativity demands you should be able to do — and the equation develops exponential instabilities. Solutions blow up. The initial-value problem becomes ill-posed. For a PDE that governs everything from heat conduction to financial derivatives, this is embarrassing.
The standard response has been to treat diffusion as an approximation that breaks down at relativistic speeds. True, but unsatisfying. The equation doesn't merely become inaccurate in the boosted frame — it becomes mathematically pathological. Even if you never approach the speed of light, the formal structure is broken.
Gavassino (arXiv:2602.21254) resolves this by finding where diffusion actually lives. Fick's law — the constitutive relation behind the diffusion equation — arises as the exact hydrodynamic sector of relativistic Fokker-Planck kinetic theory. The diffusion equation isn't fundamental. It's a projection. And the pathology comes from treating the projection as if it were the full theory.
When you embed diffusion back into its kinetic context, the instabilities vanish. The initial-value problem becomes well-posed — and not just forward in time, but backward too. The catch: only for initial conditions that admit a kinetic-theory realization. These are band-limited functions — density profiles whose spatial frequency content is bounded. The Fourier spectrum can't contain arbitrarily high frequencies because the underlying kinetic theory doesn't support them.
The Green function for this well-posed formulation is a Shannon-Whittaker interpolation kernel on full Minkowski spacetime. Obtained in closed analytic form. The solution is a discrete superposition of spatially sampled initial data, weighted by this kernel. Sampling theory applied to relativistic thermodynamics.
What makes this beautiful: the resolution isn't a patch. It's not a modified diffusion equation or an ad hoc regularization. It's the recognition that the space of physical initial conditions is smaller than the space of mathematical ones. The instability exists — but only for initial conditions that no real physical system can produce. Band-limiting is not a restriction imposed from outside. It's a consequence of the physics that diffusion approximates.
The broader lesson: when an equation breaks, the equation isn't wrong — the domain is wrong. The diffusion equation works perfectly within its natural domain of band-limited functions. It only fails when you feed it initial conditions that its parent theory can't generate. The pathology isn't in the mathematics. It's in the assumption that the mathematics applies everywhere.
Every approximation has a bandwidth. The clean ones know where their frequencies end.