Molecular Brownian motion and invariance group of the Bogolyubov   equation
  Statistics of molecular random walks in a fluid is considered with the help of Bogolyubov equation for generating functional of distribution functions. An invariance group of this equation is found. It results in many exact relations between path probability distribution of a test particle and its correlations with the fluid. As the consequence, significant restrictions on possible shape of the path distribution do arise. In particular, the hypothetical Gaussian form of long-range asymptotic proves to be forbidden, even (and first of all) under the Boltzmann-Grad limit. An allowed diffusive asymptotic possesses power-law long tail (cut off by free flight length). 
Yuriy E. Kuzovlev