Abstract:
In §§ 1–3 of the present paper we prove N. N. Bogolyubov's principle of averaging [1] for parabolic equations (theorems 1,2.2'). Lemma 2 is of most importance for the proof. Kolmogorov's theorem ([14], lemma 2.2) is essentially used for the proof of this lemma. In § 4 theorem 1 is used for studying more general parabolic and elliptic equations. The theorem of the convergence of an invariant measure of a Markov process on a torus to an invariant measure of the flow on a torus (Theorem 3) is proved in § 5.