Principle of Averaging for Parabolic and Elliptic Differential Equations and for Markov Processes with Small Diffusion

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.