  The problem of statistics of molecular random walks in a classical fluid is
analyzed by means of the BBGKY hierarchy of equations reformulated in terms of
the Bogolyubov evolution equation for generating functional of many-particle
distribution functions. A proper equivalent set of correlation functions is
introduced so that all they are integrable, vanish in statistical equilibrium,
otherwise accumulate statistical information about history of collisions of a
``molecular Brownian particle'' (test molecule) with other molecules of the
fluid. An exact evolution equation for generating functional of such
correlation functions is derived. Then it is shown that time-dependent solution
to this equation, as well as a properly defined generating functional of static
thermodynamically equilibrium correlations, possesses invariance with respect
to a definite group of transformations of independent variables of the
functional, if density of the fluid (number of molecules per unit volume) is
treated as one of the independent variables. Such invariance results in
infinitely many exact relations between the correlation functions and
probability distribution of path of the molecular Brownian particle. Even
simplest of these relations suggest significant restrictions on a profile of
the path probability distribution, even without literal solving the BBGKY
hierarchy.
