Evolution operator for the Bogolyubov (BBGKY) hierarchy. Lattice systems

The hierarchy of Bogolyubov type (BBGKY) kinetic equations for infinite classical and quantum lattice systems is considered. A formula for solving the Cauchy problem for the equations in the form $F(t)=PS(-t)F^0$ is obtained; here, $P$ is the operator of projection onto the subspace of sequences of finitely additive measures satisfying consistency conditions. Proofs are given of the uniqueness of the solution and the group property of the evolution operator in the situation when the observables are specified by uniformly continuous functions. Stationary solutions of the equations are obtained.