In a companion paper [47], we compute the full quasi-normal-mode spectrum of the VFP equation. We find that in both the Newtonian and ultrarelativistic regimes the hydrodynamic mode has the diffusive form $\omega =-i\mathfrak{D}{k}^2$, with no extra corrections. However, while in the Newtonian case this mode exists for all $k\in \mathbb{C}$, in the ultrarelativistic case it is confined to the strip $|\mathfrak{Im}k|<(2\mathfrak{D}{)}^{-1}$, ensuring that the covariant stability bound $\mathfrak{Im}\omega \le |\mathfrak{Im}k|$ [48, 49] is respected.

This implies that commonly assumed preparation protocols (such as generating a spatially localized perturbation on top of an otherwise equilibrium microscopic state) do not, in general, lie within the diffusive sector of the relativistic kinetic theory considered here.

It is worth emphasizing that, while the Vlasov-Fokker-Planck equation is a perfectly acceptable relativistic kinetic theory when regarded as a stand-alone equation, it is also a strictly Markovian dynamical model. It has been suggested that such exact Markovianity may itself be an idealization in a relativistic setting, in the sense that deriving this equation from an underlying Langevin description can require a degree of temporal or spatial nonlocality [59, 60]. On the other hand, the Langevin equation itself is subject to intrinsic limitations in relativistic contexts. Since a relativistic Hamiltonian description of interacting classical particles cannot exist [61, 62], locality arguments based on such equations are not fundamental and may be intrinsically unreliable.

Of course, the diffusion equation still defines an acausal (in the sense of Wald [64], Sec. 10.1) initial-value problem when regarded as a stand-alone partial differential equation. In this sense, diffusion is still parabolic. Nevertheless, it becomes compatible with the broader principle of relativistic causality once it is embedded into a larger microscopic theory with additional degrees of freedom, which redefine the notion of a physical signal. A closely analogous situation arises for the relativistic Schrödinger equation $i{\partial }_t\psi =\sqrt{m^2-{\partial }_x^2}\psi$, which by itself does not define a causal initial-value problem, yet corresponds to the positive-frequency sector of the Klein-Gordon equation $-{\partial }_t^2\psi =(m^2-{\partial }_x^2)\psi$, which is a perfectly causal relativistic field theory [52].