Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed

Abstract

The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion, leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix is dependent on the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentrations. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobolev-type inequality, and the Ladyzhenskaya-Ural'tseva iteration lemma.