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For classical TV-particle systems with pair interaction N" 1 X Φ(^ΐ~3j) trιe Vlasov dynamics is shown to be the w*-limit as l^i^J^N N-+QO. Propagation of molecular chaos holds in this limit, and the fluctuations of intensive observables converge to a Gaussian stochastic process.

Abstract.A new proof is given that the subtraction rules of BOGOLIUBOV and PARASIUK lead to well.defined renormalized Green's distributions. A collection of the counter-terms in "trees" removes the difficulties with overlapping divergences and allows fairly simple estimates and closed expressions for renormalized Feynman integrals. The renormalization procedure, which also applies to conventionally nonrenormalizable theories, is illustrated in the ~4-theory.

For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the quantum mechanical Hamiltonian and taken in coherent states centered in x-and p-space around h~x l2 (coordinates of a point in classical phase space) are shown to become the exponentials of coordinate functions of the classical orbit in phase space. In the same sense, h~112 [(quantum operator) (ί) -(classical function) (ί)] converges to the solution of the linear quantum mechanical system, which is obtained by linearizing the non-linear Heisenberg equations of motion around the classical orbit.

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