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We considered the qualitative behavior of the generalization of Einstein'smodel of Brownian motion when the key parameter of the time interval of\textit{free jump} degenerates. Fluids will be characterized by the number ofparticles per unit volume (density of fluid) at the point of observation.Degeneration of the phenomenon manifests in two scenarios: a) flow of thefluid, which is highly dispersing like a non-dense gas, and b) flow of fluidfar away from the source of flow, when the velocity of the flow is incomparablysmaller than the gradient of the density. First, we will show that both typesof flows can be modeled using the Einstein paradigm. We will investigate thequestion: What features will particle flow exhibit if the time interval ofthe\textit{ free jump} is inverse proportional to the density and its gradient?We will show that in this scenario, the flow exhibits localization property,namely: if at some moment of time $t_0$ in the region, the gradient of thedensity or density itself is equal to zero, then for some time $T$ during tinterval $[ t_{0}, t_0+T]$ there is no flow in the region. This directly linksto Barenblatt's finite speed of propagation property for the degenerateequation. The method of the proof is very different from Barenblatt's methodand based on the application of Ladyzhenskaya - De Giorgi iterative scheme andVespri - Tedeev technique. From PDE's point of view, it assumed that solutionexists in appropriate Sobolev type of space.

The finite speed of propagation in the degenerate Einstein-Brownian motion model